Theorem vtxdeqd 26373

Description: Equality theorem for the vertex degree: If two graphs are structurally equal, their vertex degree functions are equal. (Contributed by AV, 26-Feb-2021.)

Hypotheses

Ref	Expression
vtxdeqd.g	|- ( ph -> G e. X )
vtxdeqd.h	|- ( ph -> H e. Y )
vtxdeqd.v	|- ( ph -> (Vtx ` H ) = (Vtx ` G ) )
vtxdeqd.i	|- ( ph -> (iEdg ` H ) = (iEdg ` G ) )

Assertion

Ref	Expression
vtxdeqd	|- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Proof of Theorem vtxdeqd

Dummy variables u x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdeqd.v	. . 3 |- ( ph -> (Vtx ` H ) = (Vtx ` G ) )
2		vtxdeqd.i	. . . . . . 7 |- ( ph -> (iEdg ` H ) = (iEdg ` G ) )
3	2	dmeqd 5326	. . . . . 6 |- ( ph -> dom (iEdg ` H ) = dom (iEdg ` G ) )
4	2	fveq1d 6193	. . . . . . 7 |- ( ph -> ( (iEdg ` H ) ` x ) = ( (iEdg ` G ) ` x ) )
5	4	eleq2d 2687	. . . . . 6 |- ( ph -> ( u e. ( (iEdg ` H ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
6	3, 5	rabeqbidv 3195	. . . . 5 |- ( ph -> { x e. dom (iEdg ` H ) | u e. ( (iEdg ` H ) ` x ) } = { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } )
7	6	fveq2d 6195	. . . 4 |- ( ph -> ( # ` { x e. dom (iEdg ` H ) | u e. ( (iEdg ` H ) ` x ) } ) = ( # ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) )
8	4	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( (iEdg ` H ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
9	3, 8	rabeqbidv 3195	. . . . 5 |- ( ph -> { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } = { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } )
10	9	fveq2d 6195	. . . 4 |- ( ph -> ( # ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) = ( # ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) )
11	7, 10	oveq12d 6668	. . 3 |- ( ph -> ( ( # ` { x e. dom (iEdg ` H ) | u e. ( (iEdg ` H ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) ) = ( ( # ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) ) )
12	1, 11	mpteq12dv 4733	. 2 |- ( ph -> ( u e. (Vtx ` H ) |-> ( ( # ` { x e. dom (iEdg ` H ) | u e. ( (iEdg ` H ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) ) ) = ( u e. (Vtx ` G ) |-> ( ( # ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) ) ) )
13		vtxdeqd.h	. . 3 |- ( ph -> H e. Y )
14		eqid 2622	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
15		eqid 2622	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
16		eqid 2622	. . . 4 |- dom (iEdg ` H ) = dom (iEdg ` H )
17	14, 15, 16	vtxdgfval 26363	. . 3 |- ( H e. Y -> (VtxDeg ` H ) = ( u e. (Vtx ` H ) |-> ( ( # ` { x e. dom (iEdg ` H ) | u e. ( (iEdg ` H ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) ) ) )
18	13, 17	syl 17	. 2 |- ( ph -> (VtxDeg ` H ) = ( u e. (Vtx ` H ) |-> ( ( # ` { x e. dom (iEdg ` H ) | u e. ( (iEdg ` H ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) ) ) )
19		vtxdeqd.g	. . 3 |- ( ph -> G e. X )
20		eqid 2622	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
21		eqid 2622	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
22		eqid 2622	. . . 4 |- dom (iEdg ` G ) = dom (iEdg ` G )
23	20, 21, 22	vtxdgfval 26363	. . 3 |- ( G e. X -> (VtxDeg ` G ) = ( u e. (Vtx ` G ) |-> ( ( # ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) ) ) )
24	19, 23	syl 17	. 2 |- ( ph -> (VtxDeg ` G ) = ( u e. (Vtx ` G ) |-> ( ( # ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) +e ( # ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) ) ) )
25	12, 18, 24	3eqtr4d 2666	1 |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   {csn 4177   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   +ecxad 11944   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  VtxDegcvtxdg 26361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-vtxdg 26362

This theorem is referenced by:  eupthvdres  27095