Nuprl Lemma : pscm-id-comp
∀[C:SmallCategory]. ∀[A,B:ps_context{j:l}(C)]. ∀[sigma:psc_map{j:l}(C; A; B)].
((sigma o 1(A) = sigma ∈ psc_map{j:l}(C; A; B)) ∧ (1(B) o sigma = sigma ∈ psc_map{j:l}(C; A; B)))
Proof
Definitions occuring in Statement :
pscm-id: 1(X),
pscm-comp: G o F,
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
psc_map: A ⟶ B,
ps_context: __⊢,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x]
Lemmas referenced :
pscm-comp-sq,
pscm-id-sq,
psc_map_wf,
ps_context_wf,
small-category-cumulativity-2,
small-category_wf,
trans-id-property,
op-cat_wf,
type-cat_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
universeIsType,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
Error :memTop,
dependent_functionElimination,
productElimination,
independent_pairFormation
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[A,B:ps\_context\{j:l\}(C)]. \mforall{}[sigma:psc\_map\{j:l\}(C; A; B)].
((sigma o 1(A) = sigma) \mwedge{} (1(B) o sigma = sigma))
Date html generated:
2020_05_20-PM-01_24_17
Last ObjectModification:
2020_04_01-AM-11_00_33
Theory : presheaf!models!of!type!theory
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