### Nuprl Lemma : fpf-rename-cap2

`∀[A,C,B:Type]. ∀[eqa:EqDecider(A)]. ∀[eqc,eqc':EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B]. ∀[a:A]. ∀[z:B].`
`  rename(r;f)(r a)?z = f(a)?z ∈ B supposing Inj(A;C;r)`

Proof

Definitions occuring in Statement :  fpf-rename: `rename(r;f)` fpf-cap: `f(x)?z` fpf: `a:A fp-> B[a]` deq: `EqDecider(T)` inject: `Inj(A;B;f)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uimplies: `b supposing a` all: `∀x:A. B[x]` top: `Top` fpf-cap: `f(x)?z` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` ifthenelse: `if b then t else f fi ` bfalse: `ff` prop: `ℙ` guard: `{T}` not: `¬A` false: `False` iff: `P `⇐⇒` Q` exists: `∃x:A. B[x]` cand: `A c∧ B` inject: `Inj(A;B;f)` rev_implies: `P `` Q`
Lemmas referenced :  fpf-dom_wf subtype-fpf2 top_wf istype-void fpf-rename-ap2 equal-wf-T-base bool_wf assert_wf bnot_wf not_wf eqtt_to_assert uiff_transitivity eqff_to_assert assert_of_bnot inject_wf fpf_wf deq_wf istype-universe equal_wf fpf-rename_wf fpf-dom_functionality2 fpf-rename-dom
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity equalityTransitivity hypothesis equalitySymmetry cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality sqequalRule lambdaEquality_alt inhabitedIsType independent_isectElimination lambdaFormation_alt isect_memberEquality_alt voidElimination because_Cache baseClosed isect_memberFormation_alt unionElimination equalityElimination productElimination independent_functionElimination equalityIstype dependent_functionElimination universeIsType axiomEquality isectIsTypeImplies functionIsType instantiate universeEquality voidEquality isect_memberEquality lambdaFormation functionExtensionality lambdaEquality cumulativity hyp_replacement applyLambdaEquality productEquality independent_pairFormation dependent_pairFormation

Latex:
\mforall{}[A,C,B:Type].  \mforall{}[eqa:EqDecider(A)].  \mforall{}[eqc,eqc':EqDecider(C)].  \mforall{}[r:A  {}\mrightarrow{}  C].  \mforall{}[f:a:A  fp->  B].  \mforall{}[a:A].
\mforall{}[z:B].
rename(r;f)(r  a)?z  =  f(a)?z  supposing  Inj(A;C;r)

Date html generated: 2019_10_16-AM-11_26_15
Last ObjectModification: 2019_06_25-PM-03_26_11

Theory : finite!partial!functions

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