Theorem vtxdeqd 16220

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 4939	. . . . . 6 |- ( ph -> dom (iEdg ` H ) = dom (iEdg ` G ) )
4	2	fveq1d 5650	. . . . . . 7 |- ( ph -> ( (iEdg ` H ) ` x ) = ( (iEdg ` G ) ` x ) )
5	4	eleq2d 2301	. . . . . 6 |- ( ph -> ( u e. ( (iEdg ` H ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
6	3, 5	rabeqbidv 2798	. . . . 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 5652	. . . 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 2240	. . . . . 6 |- ( ph -> ( ( (iEdg ` H ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
9	3, 8	rabeqbidv 2798	. . . . 5 |- ( ph -> { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } = { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } )
10	9	fveq2d 5652	. . . 4 |- ( ph -> (♯ ` { x e. dom (iEdg ` H ) | ( (iEdg ` H ) ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) )
11	7, 10	oveq12d 6046	. . 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 4176	. 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 2231	. . . 4 |- (Vtx ` H ) = (Vtx ` H )
15		eqid 2231	. . . 4 |- (iEdg ` H ) = (iEdg ` H )
16		eqid 2231	. . . 4 |- dom (iEdg ` H ) = dom (iEdg ` H )
17	14, 15, 16	vtxdgfval 16212	. . 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 2231	. . . 4 |- (Vtx ` G ) = (Vtx ` G )
21		eqid 2231	. . . 4 |- (iEdg ` G ) = (iEdg ` G )
22		eqid 2231	. . . 4 |- dom (iEdg ` G ) = dom (iEdg ` G )
23	20, 21, 22	vtxdgfval 16212	. . 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 2274	1 |- ( ph -> (VtxDeg ` H ) = (VtxDeg ` G ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   {crab 2515   {csn 3673   |-> cmpt 4155   dom cdm 4731   ` cfv 5333  (class class class)co 6028   +ecxad 10049  ♯chash 11083  Vtxcvtx 15936  iEdgciedg 15937  VtxDegcvtxdg 16210

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-vtxdg 16211

This theorem is referenced by:  eupthvdres  16399