Metamath Proof Explorer


Theorem isubgr3stgrlem5

Description: Lemma 5 for isubgr3stgr . (Contributed by AV, 24-Sep-2025)

Ref Expression
Hypotheses isubgr3stgr.v V = Vtx G
isubgr3stgr.u U = G NeighbVtx X
isubgr3stgr.c C = G ClNeighbVtx X
isubgr3stgr.n N 0
isubgr3stgr.s No typesetting found for |- S = ( StarGr ` N ) with typecode |-
isubgr3stgr.w W = Vtx S
isubgr3stgr.e E = Edg G
isubgr3stgr.i I = Edg G ISubGr C
isubgr3stgr.h H = i I F i
Assertion isubgr3stgrlem5 F : C W Y I H Y = F Y

Proof

Step Hyp Ref Expression
1 isubgr3stgr.v V = Vtx G
2 isubgr3stgr.u U = G NeighbVtx X
3 isubgr3stgr.c C = G ClNeighbVtx X
4 isubgr3stgr.n N 0
5 isubgr3stgr.s Could not format S = ( StarGr ` N ) : No typesetting found for |- S = ( StarGr ` N ) with typecode |-
6 isubgr3stgr.w W = Vtx S
7 isubgr3stgr.e E = Edg G
8 isubgr3stgr.i I = Edg G ISubGr C
9 isubgr3stgr.h H = i I F i
10 9 a1i F : C W Y I H = i I F i
11 imaeq2 i = Y F i = F Y
12 11 adantl F : C W Y I i = Y F i = F Y
13 simpr F : C W Y I Y I
14 id F : C W F : C W
15 3 ovexi C V
16 15 a1i F : C W C V
17 14 16 fexd F : C W F V
18 17 adantr F : C W Y I F V
19 18 imaexd F : C W Y I F Y V
20 10 12 13 19 fvmptd F : C W Y I H Y = F Y