Nuprl Lemma : fpf-rename-cap3
∀[A,C,B:Type]. ∀[eqa:EqDecider(A)]. ∀[eqc,eqc':EqDecider(C)]. ∀[r:A ⟶ C]. ∀[f:a:A fp-> B]. ∀[a:A]. ∀[z:B]. ∀[c:C].
(rename(r;f)(c)?z = f(a)?z ∈ B) supposing ((c = (r a) ∈ C) and 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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
fpf-cap_wf,
inject_wf,
fpf_wf,
deq_wf,
fpf-rename-cap2,
fpf-rename_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
thin,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
because_Cache,
applyEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
functionEquality,
universeEquality,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination
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]. \mforall{}[c:C].
(rename(r;f)(c)?z = f(a)?z) supposing ((c = (r a)) and Inj(A;C;r))
Date html generated:
2018_05_21-PM-09_27_19
Last ObjectModification:
2018_02_09-AM-10_22_29
Theory : finite!partial!functions
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