Nuprl Lemma : cubical-snd_wf

[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ B}].  (p.2 ∈ {X ⊢ _:(B)[p.1]})


Proof




Definitions occuring in Statement :  cubical-snd: p.2 cubical-fst: p.1 cubical-sigma: Σ B csm-id-adjoin: [u] cube-context-adjoin: X.A cubical-term: {X ⊢ _:AF} csm-ap-type: (AF)s cubical-type: {X ⊢ _} cubical-set: CubicalSet uall: [x:A]. B[x] member: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T cubical-snd: p.2 cubical-term: {X ⊢ _:AF} cubical-sigma: Σ B pi1: fst(t) all: x:A. B[x] implies:  Q so_lambda: λ2x.t[x] so_apply: x[s] prop: subtype_rel: A ⊆B cubical-type: {X ⊢ _} cubical-type-at: A(a) csm-id-adjoin: [u] csm-ap-type: (AF)s csm-adjoin: (s;u) csm-ap: (s)x cc-adjoin-cube: (v;u) cubical-fst: p.1 csm-id: 1(X) top: Top type-cat: TypeCat cat-id: cat-id(C) pi2: snd(t) and: P ∧ Q cubical-type-ap-morph: (u f) identity-trans: identity-trans(C;D;F) cube-context-adjoin: X.A I-cube: X(I) functor-ob: ob(F) squash: T guard: {T} true: True uimplies: supposing a iff: ⇐⇒ Q rev_implies:  Q
Lemmas referenced :  cubical-fst_wf cubical-term_wf cubical-sigma_wf cubical-type_wf cube-context-adjoin_wf cubical-set_wf cubical-type-at_wf cc-adjoin-cube_wf pi2_wf equal_wf I-cube_wf list_wf coordinate_name_wf ident_trans_ap_lemma subtype_rel_self pi1_wf_top name-morph_wf ap_mk_nat_trans_lemma cat_id_tuple_lemma ob_pair_lemma cube-set-restriction_wf squash_wf true_wf iff_weakening_equal subtype_rel-equal csm-ap_wf csm-id-adjoin_wf subtype_rel_wf subtype_rel_weakening ext-eq_weakening csm-ap-type_wf all_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis axiomEquality equalityTransitivity equalitySymmetry isect_memberEquality because_Cache dependent_set_memberEquality setElimination rename lambdaEquality applyEquality productEquality dependent_functionElimination lambdaFormation independent_functionElimination productElimination voidElimination voidEquality independent_pairEquality promote_hyp dependent_pairEquality applyLambdaEquality imageElimination universeEquality natural_numberEquality imageMemberEquality baseClosed instantiate independent_isectElimination hyp_replacement functionEquality

Latex:
\mforall{}[X:CubicalSet].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[B:\{X.A  \mvdash{}  \_\}].  \mforall{}[p:\{X  \mvdash{}  \_:\mSigma{}  A  B\}].    (p.2  \mmember{}  \{X  \mvdash{}  \_:(B)[p.1]\})



Date html generated: 2018_05_23-PM-06_31_38
Last ObjectModification: 2018_05_20-PM-04_20_00

Theory : cubical!sets


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