Nuprl Lemma : setimages_functionality
∀b1,b2,x1,x2:coSet{i:l}. (seteq(b1;b2) ⇒ seteq(x1;x2) ⇒ seteq(setimages(b1;x1);setimages(b2;x2)))
Proof
Definitions occuring in Statement :
setimages: setimages(A;B),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
pi1: fst(t),
cand: A c∧ B,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_functionality,
exists_wf,
setmem-image,
set-image_wf,
pi1_wf,
subtype_rel_self,
subtype_rel_function,
setsubset_functionality,
setimage-iff,
setsubset_wf,
setimage_wf,
setmem-setimages-2,
seteq_weakening,
setmem_functionality,
iff_wf,
all_wf,
setmem_wf,
coSet_wf,
seteq_wf,
setimages_wf,
co-seteq-iff
Rules used in proof :
universeEquality,
promote_hyp,
functionEquality,
independent_isectElimination,
functionExtensionality,
dependent_pairEquality,
applyEquality,
spreadEquality,
dependent_pairFormation,
rename,
andLevelFunctionality,
productEquality,
impliesFunctionality,
allFunctionality,
addLevel,
lambdaEquality,
sqequalRule,
instantiate,
cumulativity,
independent_pairFormation,
because_Cache,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}b1,b2,x1,x2:coSet\{i:l\}. (seteq(b1;b2) {}\mRightarrow{} seteq(x1;x2) {}\mRightarrow{} seteq(setimages(b1;x1);setimages(b2;x2)))
Date html generated:
2018_07_29-AM-10_09_34
Last ObjectModification:
2018_07_18-PM-09_55_30
Theory : constructive!set!theory
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