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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