Nuprl Lemma : presheaf-fun-equal
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A,B:{X ⊢ _}]. ∀[f,g:{X ⊢ _:(A ⟶ B)}].
f = g ∈ {X ⊢ _:(A ⟶ B)}
supposing ∀[I:cat-ob(C)]. ∀[a:X(I)]. ∀[J:cat-ob(C)]. ∀[h:cat-arrow(C) J I]. ∀[u:A(h(a))].
((f(a) J h u) = (g(a) J h u) ∈ B(h(a)))
Proof
Definitions occuring in Statement :
presheaf-fun: (A ⟶ B),
presheaf-term-at: u(a),
presheaf-term: {X ⊢ _:A},
presheaf-type-at: A(a),
presheaf-type: {X ⊢ _},
psc-restriction: f(s),
I_set: A(I),
ps_context: __⊢,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
presheaf-term: {X ⊢ _:A},
presheaf-fun: (A ⟶ B),
all: ∀x:A. B[x],
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
squash: ↓T,
presheaf-term-at: u(a),
true: True,
implies: P ⇒ Q
Lemmas referenced :
presheaf-term_wf,
presheaf-fun_wf,
presheaf-type_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
presheaf_type_at_pair_lemma,
presheaf-type-at_wf,
psc-restriction_wf,
cat-arrow_wf,
presheaf-type-ap-morph_wf,
cat-comp_wf,
subtype_rel-equal,
psc-restriction-comp,
I_set_wf,
presheaf-term-equal,
cat-ob_wf,
presheaf-term-at_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
because_Cache,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
functionExtensionality,
setElimination,
rename,
dependent_functionElimination,
Error :memTop,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
functionIsType,
inhabitedIsType,
equalityIstype,
independent_isectElimination,
lambdaEquality_alt,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
isectIsType,
lambdaFormation_alt,
independent_functionElimination
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[f,g:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}].
f = g
supposing \mforall{}[I:cat-ob(C)]. \mforall{}[a:X(I)]. \mforall{}[J:cat-ob(C)]. \mforall{}[h:cat-arrow(C) J I]. \mforall{}[u:A(h(a))].
((f(a) J h u) = (g(a) J h u))
Date html generated:
2020_05_20-PM-01_29_41
Last ObjectModification:
2020_04_02-PM-06_26_16
Theory : presheaf!models!of!type!theory
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