Nuprl Lemma : presheaf-sigma-fun_wf
∀[C:SmallCategory]. ∀[G:ps_context{j:l}(C)]. ∀[A:{G ⊢ _}]. ∀[B,B':{G.A ⊢ _}]. ∀[f:{G.A ⊢ _:(B ⟶ B')}].
(presheaf-sigma-fun(G;A;B;f) ∈ {G ⊢ _:(Σ A B ⟶ Σ A B')})
Proof
Definitions occuring in Statement :
presheaf-sigma-fun: presheaf-sigma-fun(G;A;B;f),
presheaf-sigma: Σ A B,
presheaf-fun: (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,
all: ∀x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
presheaf-type: {X ⊢ _},
psc-snd: q,
psc-fst: p,
pscm-ap-type: (AF)s,
pscm-comp: G o F,
pscm-ap: (s)x,
compose: f o g,
presheaf-sigma-fun: presheaf-sigma-fun(G;A;B;f),
guard: {T},
uimplies: b supposing a,
pscm-adjoin: (s;u),
pscm-id-adjoin: [u],
pscm-id: 1(X),
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
true: True,
presheaf-fun: (A ⟶ B)
Lemmas referenced :
presheaf-sigma-p,
presheaf-sigma_wf,
psc-snd_wf,
presheaf-term_wf2,
small-category-cumulativity-2,
psc-adjoin_wf,
ps_context_cumulativity2,
presheaf-type-cumulativity2,
presheaf-fst_wf,
pscm-ap-type_wf,
psc-fst_wf,
pscm-adjoin_wf,
pscm-comp_wf,
presheaf-lam_wf,
subtype_rel-equal,
presheaf-fun_wf,
presheaf-type_wf,
ps_context_wf,
small-category_wf,
presheaf-pair_wf,
presheaf-app_wf_fun,
pscm-id-adjoin_wf,
presheaf-snd_wf,
pscm-ap-term_wf,
presheaf-pi_wf,
presheaf-lambda_wf,
presheaf-pi-p,
presheaf-app_wf,
pscm-presheaf-fun
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
because_Cache,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
sqequalRule,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
instantiate,
applyEquality,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[G:ps\_context\{j:l\}(C)]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[B,B':\{G.A \mvdash{} \_\}].
\mforall{}[f:\{G.A \mvdash{} \_:(B {}\mrightarrow{} B')\}].
(presheaf-sigma-fun(G;A;B;f) \mmember{} \{G \mvdash{} \_:(\mSigma{} A B {}\mrightarrow{} \mSigma{} A B')\})
Date html generated:
2020_05_20-PM-01_34_02
Last ObjectModification:
2020_04_03-AM-10_54_05
Theory : presheaf!models!of!type!theory
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