Theorem vtxdgfval 16047

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 16046	. 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 15834	. . . . 5 |- ( g e. _V -> (Vtx ` g ) e. _V )
3	2	elv 2803	. . . 4 |- (Vtx ` g ) e. _V
4		iedgex 15835	. . . . 5 |- ( g e. _V -> (iEdg ` g ) e. _V )
5	4	elv 2803	. . . 4 |- (iEdg ` g ) e. _V
6		simpl 109	. . . . 5 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> v = (Vtx ` g ) )
7		dmeq 4923	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> dom e = dom (iEdg ` g ) )
8		fveq1 5628	. . . . . . . . . 10 |- ( e = (iEdg ` g ) -> ( e ` x ) = ( (iEdg ` g ) ` x ) )
9	8	eleq2d 2299	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( u e. ( e ` x ) <-> u e. ( (iEdg ` g ) ` x ) ) )
10	7, 9	rabeqbidv 2794	. . . . . . . 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 5633	. . . . . . 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 2238	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( ( e ` x ) = { u } <-> ( (iEdg ` g ) ` x ) = { u } ) )
13	7, 12	rabeqbidv 2794	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | ( e ` x ) = { u } } = { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } )
14	13	fveq2d 5633	. . . . . . 7 |- ( e = (iEdg ` g ) -> (♯ ` { x e. dom e | ( e ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) )
15	11, 14	oveq12d 6025	. . . . . 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 4166	. . . 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 3174	. . 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 5629	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
20		vtxdgfval.v	. . . . . 6 |- V = (Vtx ` G )
21	19, 20	eqtr4di 2280	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
22		fveq2 5629	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
23	22	dmeqd 4925	. . . . . . . . 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 4924	. . . . . . . . . 10 |- dom I = dom (iEdg ` G )
27	24, 26	eqtri 2250	. . . . . . . . 9 |- A = dom (iEdg ` G )
28	23, 27	eqtr4di 2280	. . . . . . . 8 |- ( g = G -> dom (iEdg ` g ) = A )
29	22, 25	eqtr4di 2280	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = I )
30	29	fveq1d 5631	. . . . . . . . 9 |- ( g = G -> ( (iEdg ` g ) ` x ) = ( I ` x ) )
31	30	eleq2d 2299	. . . . . . . 8 |- ( g = G -> ( u e. ( (iEdg ` g ) ` x ) <-> u e. ( I ` x ) ) )
32	28, 31	rabeqbidv 2794	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } = { x e. A | u e. ( I ` x ) } )
33	32	fveq2d 5633	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) = (♯ ` { x e. A | u e. ( I ` x ) } ) )
34	30	eqeq1d 2238	. . . . . . . 8 |- ( g = G -> ( ( (iEdg ` g ) ` x ) = { u } <-> ( I ` x ) = { u } ) )
35	28, 34	rabeqbidv 2794	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } = { x e. A | ( I ` x ) = { u } } )
36	35	fveq2d 5633	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) = (♯ ` { x e. A | ( I ` x ) = { u } } ) )
37	33, 36	oveq12d 6025	. . . . 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 4166	. . . 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 2274	. 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 2811	. 2 |- ( G e. W -> G e. _V )
42		vtxex 15834	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
43	20, 42	eqeltrid 2316	. . 3 |- ( G e. W -> V e. _V )
44	43	mptexd 5870	. 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 5718	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 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   [_csb 3124   {csn 3666   |-> 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:  vtxdgfifival  16050  vtxdgop  16051  vtxdgfif  16052  vtxdeqd  16055