Nuprl Lemma : A-face-image_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[I,K:Cname List]. ∀[f:name-morph(I;K)]. ∀[alpha:X(I)]. ∀[face:A-face(X;A;I;alpha)].
A-face-image(X;A;I;K;f;alpha;face) ∈ A-face(X;A;K;f(alpha)) supposing ↑isname(f (fst(face)))
Proof
Definitions occuring in Statement :
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
A-face: A-face(X;A;I;alpha),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
isname: isname(z),
coordinate_name: Cname,
list: T List,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
member: t ∈ T,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
A-face: A-face(X;A;I;alpha),
spreadn: spread3,
pi1: fst(t),
nameset: nameset(L),
name-morph: name-morph(I;J),
prop: ℙ,
uiff: uiff(P;Q),
and: P ∧ Q,
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
true: True,
squash: ↓T,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
assert-isname,
l_member_wf,
coordinate_name_wf,
name-morph_subtype_remove1,
cubical-type-ap-morph_wf,
cube-set-restriction_wf,
cubical-type-at_wf,
face-map_wf2,
int_seg_wf,
list-diff_wf,
cons_wf,
assert_wf,
isname_wf,
pi1_wf_top,
nameset_wf,
A-face_wf,
I-cube_wf,
name-morph_wf,
list_wf,
cubical-type_wf,
cubical-set_wf,
cname_deq_wf,
nil_wf,
equal_wf,
name-comp_wf,
squash_wf,
true_wf,
face-map-comp,
iff_weakening_equal,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
setElimination,
rename,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
dependent_set_memberEquality,
independent_isectElimination,
dependent_pairEquality,
productEquality,
natural_numberEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
instantiate,
imageElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I,K:Cname List]. \mforall{}[f:name-morph(I;K)]. \mforall{}[alpha:X(I)].
\mforall{}[face:A-face(X;A;I;alpha)].
A-face-image(X;A;I;K;f;alpha;face) \mmember{} A-face(X;A;K;f(alpha)) supposing \muparrow{}isname(f (fst(face)))
Date html generated:
2017_10_05-AM-10_18_07
Last ObjectModification:
2017_07_28-AM-11_20_36
Theory : cubical!sets
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