Theorem vtxdeqd 16451

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 4978	. . . . . 6 |- ( ph -> dom (iEdg ` H ) = dom (iEdg ` G ) )
4	2	fveq1d 5692	. . . . . . 7 |- ( ph -> ( (iEdg ` H ) ` x ) = ( (iEdg ` G ) ` x ) )
5	4	eleq2d 2308	. . . . . 6 |- ( ph -> ( u e. ( (iEdg ` H ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
6	3, 5	rabeqbidv 2816	. . . . 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 5694	. . . 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 2247	. . . . . 6 |- ( ph -> ( ( (iEdg ` H ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
9	3, 8	rabeqbidv 2816	. . . . 5 |- ( ph -> { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } = { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } )
10	9	fveq2d 5694	. . . 4 |- ( ph -> (♯ ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) )
11	7, 10	oveq12d 6093	. . 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 4208	. 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 2238	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
15		eqid 2238	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
16		eqid 2238	. . . 4 |- dom (iEdg ` H ) = dom (iEdg ` H )
17	14, 15, 16	vtxdgfval 16443	. . 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 2238	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
21		eqid 2238	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
22		eqid 2238	. . . 4 |- dom (iEdg ` G ) = dom (iEdg ` G )
23	20, 21, 22	vtxdgfval 16443	. . 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 2281	1 |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   {csn 3705   |-> cmpt 4187   dom cdm 4769   ` cfv 5372  (class class class)co 6075   +ecxad 10151  ♯chash 11192  Vtxcvtx 16167  iEdgciedg 16168  VtxDegcvtxdg 16441

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-vtxdg 16442

This theorem is referenced by:  eupthvdres  16630