Nuprl Lemma : cat-isomorphic_weakening
∀C:SmallCategory. ∀x,y:cat-ob(C). cat-isomorphic(C;x;y) supposing x = y ∈ cat-ob(C)
Proof
Definitions occuring in Statement :
cat-isomorphic: cat-isomorphic(C;x;y),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uimplies: b supposing a,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
cat-isomorphic: cat-isomorphic(C;x;y),
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cat-id_wf,
subtype_rel_self,
cat-arrow_wf,
subtype_rel_wf,
cat-isomorphism_wf,
squash_wf,
true_wf,
iff_weakening_equal,
cat-id-isomorphism,
equal_wf,
cat-ob_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
dependent_pairFormation,
applyEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
sqequalRule,
lambdaEquality,
imageElimination,
equalityTransitivity,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}C:SmallCategory. \mforall{}x,y:cat-ob(C). cat-isomorphic(C;x;y) supposing x = y
Date html generated:
2020_05_20-AM-07_50_17
Last ObjectModification:
2017_01_10-PM-06_12_06
Theory : small!categories
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