Nuprl Lemma : free-from-atom-fpf-ap

[a:Atom1]. ∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ 𝕌']. ∀[f:x:A fp-> B[x]].
  ∀[x:A]. (a#f(x):B[x]) supposing ((↑x ∈ dom(f)) and a#x:A) supposing a#f:x:A fp-> B[x]


Definitions occuring in Statement :  fpf-ap: f(x) fpf-dom: x ∈ dom(f) fpf: a:A fp-> B[a] deq: EqDecider(T) free-from-atom: a#x:T atom: Atom$n assert: b uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] uimplies: supposing a member: t ∈ T so_apply: x[s] so_lambda: λ2x.t[x] sq_stable: SqStable(P) implies:  Q all: x:A. B[x] subtype_rel: A ⊆B top: Top iff: ⇐⇒ Q and: P ∧ Q squash: T prop: true: True fpf-ap: f(x) fpf-domain: fpf-domain(f) fpf: a:A fp-> B[a]
Lemmas referenced :  sq_stable__free-from-atom fpf-ap_wf member-fpf-domain subtype-fpf2 top_wf assert_wf fpf-dom_wf free-from-atom_wf fpf_wf deq_wf l_member_wf fpf-domain_wf set_wf equal_wf pi2_wf list_wf pi1_wf_top
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation cut thin instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination applyEquality hypothesisEquality cumulativity sqequalRule lambdaEquality independent_isectElimination hypothesis independent_functionElimination because_Cache dependent_functionElimination lambdaFormation isect_memberEquality voidElimination voidEquality productElimination imageMemberEquality baseClosed imageElimination functionEquality universeEquality atomnEquality dependent_set_memberEquality freeFromAtomApplication freeFromAtomSet hyp_replacement equalitySymmetry setElimination rename setEquality natural_numberEquality equalityTransitivity freeFromAtomTriviality independent_pairEquality productEquality

\mforall{}[a:Atom1].  \mforall{}[A:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[B:A  {}\mrightarrow{}  \mBbbU{}'].  \mforall{}[f:x:A  fp->  B[x]].
    \mforall{}[x:A].  (a\#f(x):B[x])  supposing  ((\muparrow{}x  \mmember{}  dom(f))  and  a\#x:A)  supposing  a\#f:x:A  fp->  B[x]

Date html generated: 2019_10_16-AM-11_26_47
Last ObjectModification: 2018_09_18-PM-10_17_58

Theory : finite!partial!functions

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