Theorem vtxdgop 16142

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 15868	. . . . 5 |- ( G e. W -> (Vtx ` G ) e. _V )
2		iedgex 15869	. . . . 5 |- ( G e. W -> (iEdg ` G ) e. _V )
3		opexg 4320	. . . . 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 2231	. . . . 5 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
6		eqid 2231	. . . . 5 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
7		eqid 2231	. . . . 5 |- dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
8	5, 6, 7	vtxdgfval 16138	. . . 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 15872	. . . . 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 15875	. . . . . . . . 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 4933	. . . . . . 7 |- ( G e. W -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
15	13	fveq1d 5641	. . . . . . . 8 |- ( G e. W -> ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = ( (iEdg ` G ) ` x ) )
16	15	eleq2d 2301	. . . . . . 7 |- ( G e. W -> ( u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
17	14, 16	rabeqbidv 2797	. . . . . 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 5643	. . . . 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 2240	. . . . . . 7 |- ( G e. W -> ( ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
20	14, 19	rabeqbidv 2797	. . . . . 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 5643	. . . . 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 6035	. . . 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 4171	. . 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 2264	. 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 6020	. . 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 2231	. . 3 |- (Vtx ` G ) = (Vtx ` G )
28		eqid 2231	. . 3 |- (iEdg ` G ) = (iEdg ` G )
29		eqid 2231	. . 3 |- dom (iEdg ` G ) = dom (iEdg ` G )
30	27, 28, 29	vtxdgfval 16138	. 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 2275	1 |- ( G e. W -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {crab 2514   _Vcvv 2802   {csn 3669   <.cop 3672   |-> cmpt 4150   dom cdm 4725   ` cfv 5326  (class class class)co 6017   +ecxad 10004  ♯chash 11036  Vtxcvtx 15862  iEdgciedg 15863  VtxDegcvtxdg 16136

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-vtxdg 16137

This theorem is referenced by: (None)