Nuprl Lemma : cat-initial-isomorphic
∀[C:SmallCategory]. ∀i1,i2:cat-ob(C). (Initial(i1) ⇒ Initial(i2) ⇒ cat-isomorphic(C;i1;i2))
Proof
Definitions occuring in Statement :
cat-initial: Initial(i),
cat-isomorphic: cat-isomorphic(C;x;y),
cat-ob: cat-ob(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
cat-initial: Initial(i),
member: t ∈ T,
and: P ∧ Q,
cat-isomorphic: cat-isomorphic(C;x;y),
exists: ∃x:A. B[x],
prop: ℙ,
cat-isomorphism: cat-isomorphism(C;x;y;f),
cand: A c∧ B,
cat-inverse: fg=1,
guard: {T}
Lemmas referenced :
cat-isomorphism_wf,
cat-initial_wf,
cat-ob_wf,
small-category_wf,
cat-inverse_wf,
cat-comp_wf,
cat-id_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
rename,
dependent_pairFormation,
introduction,
extract_by_obid,
levelHypothesis,
independent_pairFormation,
productEquality,
applyEquality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}i1,i2:cat-ob(C). (Initial(i1) {}\mRightarrow{} Initial(i2) {}\mRightarrow{} cat-isomorphic(C;i1;i2))
Date html generated:
2017_01_10-AM-08_40_50
Last ObjectModification:
2017_01_09-AM-10_03_27
Theory : small!categories
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