### Nuprl Lemma : fpf-all_wf

`∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[P:x:{x:A| ↑x ∈ dom(f)}  ⟶ B[x] ⟶ ℙ].`
`  (∀x∈dom(f). v=f(x) ``  P[x;v] ∈ ℙ)`

Proof

Definitions occuring in Statement :  fpf-all: `∀x∈dom(f). v=f(x) ``  P[x; v]` fpf-dom: `x ∈ dom(f)` fpf: `a:A fp-> B[a]` deq: `EqDecider(T)` assert: `↑b` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` member: `t ∈ T` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  fpf-all: `∀x∈dom(f). v=f(x) ``  P[x; v]` uall: `∀[x:A]. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` implies: `P `` Q` prop: `ℙ` subtype_rel: `A ⊆r B` so_apply: `x[s]` uimplies: `b supposing a` all: `∀x:A. B[x]` top: `Top` so_apply: `x[s1;s2]`
Lemmas referenced :  all_wf assert_wf fpf-dom_wf subtype-fpf2 top_wf fpf-ap_wf fpf_wf deq_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality lambdaEquality functionEquality because_Cache applyEquality hypothesis independent_isectElimination lambdaFormation isect_memberEquality voidElimination voidEquality dependent_set_memberEquality universeEquality axiomEquality equalityTransitivity equalitySymmetry setEquality setElimination rename

Latex:
\mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[f:x:A  fp->  B[x]].  \mforall{}[P:x:\{x:A|  \muparrow{}x  \mmember{}  dom(f)\}
{}\mrightarrow{}  B[x]
{}\mrightarrow{}  \mBbbP{}].
(\mforall{}x\mmember{}dom(f).  v=f(x)  {}\mRightarrow{}    P[x;v]  \mmember{}  \mBbbP{})

Date html generated: 2018_05_21-PM-09_26_23
Last ObjectModification: 2018_02_09-AM-10_21_51

Theory : finite!partial!functions

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