Theorem vtxdgfval 16283

Description: The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 9-Dec-2020.)

Hypotheses

Ref	Expression
vtxdgfval.v	|- V = (Vtx ` G )
vtxdgfval.i	|- I = (iEdg ` G )
vtxdgfval.a	|- A = dom I

Assertion

Ref	Expression
vtxdgfval	|- ( G e. W -> (VtxDeg ` G ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )

Distinct variable groups:   x, u   x, A   u, G, x   u, V

Allowed substitution hints:   A( u)   I( x, u)   V( x)   W( x, u)

Proof of Theorem vtxdgfval

Dummy variables e g v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vtxdg 16282	. 2 |- VtxDeg = ( g e. _V |-> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) ) )
2		vtxex 16013	. . . . 5 |- ( g e. _V -> (Vtx ` g ) e. _V )
3	2	elv 2817	. . . 4 |- (Vtx ` g ) e. _V
4		iedgex 16014	. . . . 5 |- ( g e. _V -> (iEdg ` g ) e. _V )
5	4	elv 2817	. . . 4 |- (iEdg ` g ) e. _V
6		simpl 109	. . . . 5 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> v = (Vtx ` g ) )
7		dmeq 4956	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> dom e = dom (iEdg ` g ) )
8		fveq1 5669	. . . . . . . . . 10 |- ( e = (iEdg ` g ) -> ( e ` x ) = ( (iEdg ` g ) ` x ) )
9	8	eleq2d 2302	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( u e. ( e ` x ) <-> u e. ( (iEdg ` g ) ` x ) ) )
10	7, 9	rabeqbidv 2808	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | u e. ( e ` x ) } = { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } )
11	10	fveq2d 5674	. . . . . . 7 |- ( e = (iEdg ` g ) -> (♯ ` { x e. dom e | u e. ( e ` x ) } ) = (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) )
12	8	eqeq1d 2241	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( ( e ` x ) = { u } <-> ( (iEdg ` g ) ` x ) = { u } ) )
13	7, 12	rabeqbidv 2808	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | ( e ` x ) = { u } } = { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } )
14	13	fveq2d 5674	. . . . . . 7 |- ( e = (iEdg ` g ) -> (♯ ` { x e. dom e | ( e ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) )
15	11, 14	oveq12d 6068	. . . . . 6 |- ( e = (iEdg ` g ) -> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) = ( (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) ) )
16	15	adantl 277	. . . . 5 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) = ( (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) ) )
17	6, 16	mpteq12dv 4192	. . . 4 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` 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 } } ) ) ) )
18	3, 5, 17	csbie2 3188	. . 3 |- [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` 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 } } ) ) )
19		fveq2 5670	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
20		vtxdgfval.v	. . . . . 6 |- V = (Vtx ` G )
21	19, 20	eqtr4di 2283	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
22		fveq2 5670	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
23	22	dmeqd 4958	. . . . . . . . 9 |- ( g = G -> dom (iEdg ` g ) = dom (iEdg ` G ) )
24		vtxdgfval.a	. . . . . . . . . 10 |- A = dom I
25		vtxdgfval.i	. . . . . . . . . . 11 |- I = (iEdg ` G )
26	25	dmeqi 4957	. . . . . . . . . 10 |- dom I = dom (iEdg ` G )
27	24, 26	eqtri 2253	. . . . . . . . 9 |- A = dom (iEdg ` G )
28	23, 27	eqtr4di 2283	. . . . . . . 8 |- ( g = G -> dom (iEdg ` g ) = A )
29	22, 25	eqtr4di 2283	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = I )
30	29	fveq1d 5672	. . . . . . . . 9 |- ( g = G -> ( (iEdg ` g ) ` x ) = ( I ` x ) )
31	30	eleq2d 2302	. . . . . . . 8 |- ( g = G -> ( u e. ( (iEdg ` g ) ` x ) <-> u e. ( I ` x ) ) )
32	28, 31	rabeqbidv 2808	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } = { x e. A | u e. ( I ` x ) } )
33	32	fveq2d 5674	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) = (♯ ` { x e. A | u e. ( I ` x ) } ) )
34	30	eqeq1d 2241	. . . . . . . 8 |- ( g = G -> ( ( (iEdg ` g ) ` x ) = { u } <-> ( I ` x ) = { u } ) )
35	28, 34	rabeqbidv 2808	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } = { x e. A | ( I ` x ) = { u } } )
36	35	fveq2d 5674	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) = (♯ ` { x e. A | ( I ` x ) = { u } } ) )
37	33, 36	oveq12d 6068	. . . . 5 |- ( g = G -> ( (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) +e (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) ) = ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) )
38	21, 37	mpteq12dv 4192	. . . 4 |- ( g = 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 } } ) ) ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )
39	38	adantl 277	. . 3 |- ( ( G e. W /\ g = 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 } } ) ) ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )
40	18, 39	eqtrid 2277	. 2 |- ( ( G e. W /\ g = G ) -> [_ (Vtx ` g ) / v ]_ [_ (iEdg ` g ) / e ]_ ( u e. v |-> ( (♯ ` { x e. dom e | u e. ( e ` x ) } ) +e (♯ ` { x e. dom e | ( e ` x ) = { u } } ) ) ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )
41		elex 2825	. 2 |- ( G e. W -> G e. _V )
42		vtxex 16013	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
43	20, 42	eqeltrid 2319	. . 3 |- ( G e. W -> V e. _V )
44	43	mptexd 5913	. 2 |- ( G e. W -> ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) e. _V )
45	1, 40, 41, 44	fvmptd2 5759	1 |- ( G e. W -> (VtxDeg ` G ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {crab 2524   _Vcvv 2813   [_csb 3138   {csn 3689   |-> cmpt 4171   dom cdm 4749   ` cfv 5352  (class class class)co 6050   +ecxad 10103  ♯chash 11138  Vtxcvtx 16007  iEdgciedg 16008  VtxDegcvtxdg 16281

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-vtxdg 16282

This theorem is referenced by:  vtxdgfifival  16286  vtxdgop  16287  vtxdgfif  16288  vtxdeqd  16291