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Step * of 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)))
BY
{ (Intros
   THEN Unhide
   THEN Unfold `A-face-image` 0
   THEN DVar `face'
   THEN DVar `f1'
   THEN Reduce 0
   THEN Unfold `A-face` 0
   THEN Reduce (-1)
   THEN DVar `x'
   THEN (FLemma `assert-isname` [-1] THENA Auto)
   THEN (Assert ⌜f ∈ name-morph(I-[x];K-[f x])⌝⋅ THENA (BLemma `name-morph_subtype_remove1` ⋅ THEN Auto))
   THEN (Assert ⌜A(f((x:=i)(alpha))) = A((f x:=i)(f(alpha))) ∈ Type⌝⋅
         THENA (RWO "cube-set-restriction-comp" 0 THEN Auto)
         )
   THEN Auto) }

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Latex:

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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)))


By

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Latex:
(Intros
  THEN  Unhide
  THEN  Unfold  `A-face-image`  0
  THEN  DVar  `face'
  THEN  DVar  `f1'
  THEN  Reduce  0
  THEN  Unfold  `A-face`  0
  THEN  Reduce  (-1)
  THEN  DVar  `x'
  THEN  (FLemma  `assert-isname`  [-1]  THENA  Auto)
  THEN  (Assert  \mkleeneopen{}f  \mmember{}  name-morph(I-[x];K-[f  x])\mkleeneclose{}\mcdot{}
              THENA  (BLemma  `name-morph\_subtype\_remove1`  \mcdot{}  THEN  Auto)
              )
  THEN  (Assert  \mkleeneopen{}A(f((x:=i)(alpha)))  =  A((f  x:=i)(f(alpha)))\mkleeneclose{}\mcdot{}
              THENA  (RWO  "cube-set-restriction-comp"  0  THEN  Auto)
              )
  THEN  Auto)

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