Nuprl Lemma : op-op-cat
∀[C:SmallCategory]. (op-cat(op-cat(C)) = C ∈ SmallCategory)
Proof
Definitions occuring in Statement :
op-cat: op-cat(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
small-category: SmallCategory,
member: t ∈ T,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
spreadn: spread4,
op-cat: op-cat(C),
all: ∀x:A. B[x]
Lemmas referenced :
small-category_wf,
iff_weakening_equal,
eta_conv,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
universeIsType,
introduction,
extract_by_obid,
productEquality,
independent_pairEquality,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
cumulativity,
functionEquality,
universeEquality,
equalityTransitivity,
isectElimination,
imageElimination,
lambdaEquality,
instantiate,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_pairEquality,
sqequalRule,
productElimination,
productIsType,
functionIsType,
equalityIsType1
Latex:
\mforall{}[C:SmallCategory]. (op-cat(op-cat(C)) = C)
Date html generated:
2019_10_31-AM-07_24_06
Last ObjectModification:
2018_11_10-AM-11_32_30
Theory : small!categories
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