Theorem vtxdgop 16051

Description: The vertex degree expressed as operation. (Contributed by AV, 12-Dec-2021.)

Assertion

Ref	Expression
vtxdgop	|- ( G e. W -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )

Proof of Theorem vtxdgop

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

Step	Hyp	Ref	Expression
1		vtxex 15834	. . . . 5 |- ( G e. W -> (Vtx ` G ) e. _V )
2		iedgex 15835	. . . . 5 |- ( G e. W -> (iEdg ` G ) e. _V )
3		opexg 4314	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> <. (Vtx ` G ) , (iEdg ` G ) >. e. _V )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( G e. W -> <. (Vtx ` G ) , (iEdg ` G ) >. e. _V )
5		eqid 2229	. . . . 5 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
6		eqid 2229	. . . . 5 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
7		eqid 2229	. . . . 5 |- dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
8	5, 6, 7	vtxdgfval 16047	. . . 4 |- ( <. (Vtx ` G ) , (iEdg ` G ) >. e. _V -> (VtxDeg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = ( u e. (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) |-> ( (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } } ) ) ) )
9	4, 8	syl 14	. . 3 |- ( G e. W -> (VtxDeg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = ( u e. (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) |-> ( (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } } ) ) ) )
10		opvtxfv 15838	. . . . 5 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G ) )
11	1, 2, 10	syl2anc 411	. . . 4 |- ( G e. W -> (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` G ) )
12		opiedgfv 15841	. . . . . . . . 9 |- ( ( (Vtx ` G ) e. _V /\ (iEdg ` G ) e. _V ) -> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G ) )
13	1, 2, 12	syl2anc 411	. . . . . . . 8 |- ( G e. W -> (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` G ) )
14	13	dmeqd 4925	. . . . . . 7 |- ( G e. W -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
15	13	fveq1d 5631	. . . . . . . 8 |- ( G e. W -> ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = ( (iEdg ` G ) ` x ) )
16	15	eleq2d 2299	. . . . . . 7 |- ( G e. W -> ( u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
17	14, 16	rabeqbidv 2794	. . . . . 6 |- ( G e. W -> { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) } = { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } )
18	17	fveq2d 5633	. . . . 5 |- ( G e. W -> (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) } ) = (♯ ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) )
19	15	eqeq1d 2238	. . . . . . 7 |- ( G e. W -> ( ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
20	14, 19	rabeqbidv 2794	. . . . . 6 |- ( G e. W -> { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } } = { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } )
21	20	fveq2d 5633	. . . . 5 |- ( G e. W -> (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) )
22	18, 21	oveq12d 6025	. . . 4 |- ( G e. W -> ( (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } } ) ) = ( (♯ ` { x e. dom (iEdg ` G ) | u e. ( (iEdg ` G ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` G ) | ( (iEdg ` G ) ` x ) = { u } } ) ) )
23	11, 22	mpteq12dv 4166	. . 3 |- ( G e. W -> ( u e. (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) |-> ( (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) | ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` 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 } } ) ) ) )
24	9, 23	eqtrd 2262	. 2 |- ( G e. W -> (VtxDeg ` <. (Vtx ` G ) , (iEdg ` 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		df-ov 6010	. . 3 |- ( (Vtx ` G )VtxDeg (iEdg ` G ) ) = (VtxDeg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
26	25	a1i 9	. 2 |- ( G e. W -> ( (Vtx ` G )VtxDeg (iEdg ` G ) ) = (VtxDeg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) )
27		eqid 2229	. . 3 |- (Vtx ` G ) = (Vtx ` G )
28		eqid 2229	. . 3 |- (iEdg ` G ) = (iEdg ` G )
29		eqid 2229	. . 3 |- dom (iEdg ` G ) = dom (iEdg ` G )
30	27, 28, 29	vtxdgfval 16047	. 2 |- ( G e. W -> (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 } } ) ) ) )
31	24, 26, 30	3eqtr4rd 2273	1 |- ( G e. W -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   {csn 3666   <.cop 3669   |-> 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)