Theorem vtxdgop 16413

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 16139	. . . . 5 |- ( G e. W -> (Vtx ` G ) e. _V )
2		iedgex 16140	. . . . 5 |- ( G e. W -> (iEdg ` G ) e. _V )
3		opexg 4349	. . . . 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 2234	. . . . 5 |- (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (Vtx ` <. (Vtx ` G ) , (iEdg ` G ) >. )
6		eqid 2234	. . . . 5 |- (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
7		eqid 2234	. . . . 5 |- dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. )
8	5, 6, 7	vtxdgfval 16409	. . . 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 16143	. . . . 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 16146	. . . . . . . . 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 4963	. . . . . . 7 |- ( G e. W -> dom (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) = dom (iEdg ` G ) )
15	13	fveq1d 5677	. . . . . . . 8 |- ( G e. W -> ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = ( (iEdg ` G ) ` x ) )
16	15	eleq2d 2304	. . . . . . 7 |- ( G e. W -> ( u e. ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) <-> u e. ( (iEdg ` G ) ` x ) ) )
17	14, 16	rabeqbidv 2810	. . . . . 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 5679	. . . . 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 2243	. . . . . . 7 |- ( G e. W -> ( ( (iEdg ` <. (Vtx ` G ) , (iEdg ` G ) >. ) ` x ) = { u } <-> ( (iEdg ` G ) ` x ) = { u } ) )
20	14, 19	rabeqbidv 2810	. . . . . 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 5679	. . . . 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 6076	. . . 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 4197	. . 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 2267	. 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 6061	. . 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 2234	. . 3 |- (Vtx ` G ) = (Vtx ` G )
28		eqid 2234	. . 3 |- (iEdg ` G ) = (iEdg ` G )
29		eqid 2234	. . 3 |- dom (iEdg ` G ) = dom (iEdg ` G )
30	27, 28, 29	vtxdgfval 16409	. 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 2278	1 |- ( G e. W -> (VtxDeg ` G ) = ( (Vtx ` G )VtxDeg (iEdg ` G ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {crab 2526   _Vcvv 2815   {csn 3694   <.cop 3697   |-> cmpt 4176   dom cdm 4754   ` cfv 5357  (class class class)co 6058   +ecxad 10122  ♯chash 11163  Vtxcvtx 16133  iEdgciedg 16134  VtxDegcvtxdg 16407

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-vtxdg 16408

This theorem is referenced by: (None)