Nuprl Lemma : presheaf-fst_wf
∀[C:SmallCategory]. ∀[X:ps_context{j:l}(C)]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. (p.1 ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
presheaf-fst: p.1,
presheaf-sigma: Σ A B,
psc-adjoin: X.A,
presheaf-term: {X ⊢ _:A},
presheaf-type: {X ⊢ _},
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-fst: p.1,
presheaf-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
presheaf-sigma: Σ A B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi1: fst(t),
presheaf-type: {X ⊢ _},
and: P ∧ Q,
presheaf-type-ap-morph: (u a f),
presheaf-type-at: A(a),
pi2: snd(t),
prop: ℙ
Lemmas referenced :
presheaf-term_wf,
presheaf-sigma_wf,
presheaf-type_wf,
psc-adjoin_wf,
small-category-cumulativity-2,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
ps_context_wf,
small-category_wf,
presheaf_type_at_pair_lemma,
I_set_wf,
cat-ob_wf,
presheaf_type_ap_morph_pair_lemma,
cat-arrow_wf,
equal_wf,
psc-restriction_wf,
pi1_wf_top
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
dependent_functionElimination,
Error :memTop,
lambdaEquality_alt,
lambdaFormation_alt,
productElimination,
equalityIstype,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
because_Cache,
independent_pairEquality,
functionIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:ps\_context\{j:l\}(C)]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[p:\{X \mvdash{} \_:\mSigma{} A B\}].
(p.1 \mmember{} \{X \mvdash{} \_:A\})
Date html generated:
2020_05_20-PM-01_31_43
Last ObjectModification:
2020_04_02-PM-03_04_38
Theory : presheaf!models!of!type!theory
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