Nuprl Lemma : psc-restriction-comp2
∀C:SmallCategory. ∀X:ps_context{j:l}(C). ∀I,J1,J2,K:cat-ob(C). ∀f:cat-arrow(C) J1 I. ∀g:cat-arrow(C) K J1. ∀a:X(I).
g(f(a)) = cat-comp(C) K J1 I g f(a) ∈ X(K) supposing J1 = J2 ∈ cat-ob(C)
Proof
Definitions occuring in Statement :
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
I_set_wf,
psc-restriction_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
subtype_rel-equal,
cat-arrow_wf,
cat-comp_wf,
iff_weakening_equal,
psc-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
independent_isectElimination,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
universeIsType
Latex:
\mforall{}C:SmallCategory. \mforall{}X:ps\_context\{j:l\}(C). \mforall{}I,J1,J2,K:cat-ob(C). \mforall{}f:cat-arrow(C) J1 I.
\mforall{}g:cat-arrow(C) K J1. \mforall{}a:X(I).
g(f(a)) = cat-comp(C) K J1 I g f(a) supposing J1 = J2
Date html generated:
2020_05_20-PM-01_24_33
Last ObjectModification:
2020_04_01-AM-11_00_38
Theory : presheaf!models!of!type!theory
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