Theorem vtxdeqd 16055

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 4925	. . . . . 6 |- ( ph -> dom (iEdg ` H ) = dom (iEdg ` G ) )
4	2	fveq1d 5631	. . . . . . 7 |- ( ph -> ( (iEdg ` H ) ` x ) = ( (iEdg ` G ) ` x ) )
5	4	eleq2d 2299	. . . . . 6 |- ( ph -> ( u e. ( (iEdg ` H ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
6	3, 5	rabeqbidv 2794	. . . . 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 5633	. . . 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 2238	. . . . . 6 |- ( ph -> ( ( (iEdg ` H ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
9	3, 8	rabeqbidv 2794	. . . . 5 |- ( ph -> { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } = { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } )
10	9	fveq2d 5633	. . . 4 |- ( ph -> (♯ ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) )
11	7, 10	oveq12d 6025	. . 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 4166	. 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 2229	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
15		eqid 2229	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
16		eqid 2229	. . . 4 |- dom (iEdg ` H ) = dom (iEdg ` H )
17	14, 15, 16	vtxdgfval 16047	. . 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 14	. 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 2229	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
21		eqid 2229	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
22		eqid 2229	. . . 4 |- dom (iEdg ` G ) = dom (iEdg ` G )
23	20, 21, 22	vtxdgfval 16047	. . 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 14	. 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 2272	1 |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   {csn 3666   |-> cmpt 4145   dom cdm 4719   ` cfv 5318  (class class class)co 6007   +ecxad 9978  ♯chash 11009  Vtxcvtx 15828  iEdgciedg 15829  VtxDegcvtxdg 16045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-vtxdg 16046

This theorem is referenced by: (None)