Theorem vtxdgfval 16409

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 16408	. 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 16139	. . . . 5 |- ( g e. _V -> (Vtx ` g ) e. _V )
3	2	elv 2819	. . . 4 |- (Vtx ` g ) e. _V
4		iedgex 16140	. . . . 5 |- ( g e. _V -> (iEdg ` g ) e. _V )
5	4	elv 2819	. . . 4 |- (iEdg ` g ) e. _V
6		simpl 109	. . . . 5 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> v = (Vtx ` g ) )
7		dmeq 4961	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> dom e = dom (iEdg ` g ) )
8		fveq1 5674	. . . . . . . . . 10 |- ( e = (iEdg ` g ) -> ( e ` x ) = ( (iEdg ` g ) ` x ) )
9	8	eleq2d 2304	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( u e. ( e ` x ) <-> u e. ( (iEdg ` g ) ` x ) ) )
10	7, 9	rabeqbidv 2810	. . . . . . . 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 5679	. . . . . . 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 2243	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( ( e ` x ) = { u } <-> ( (iEdg ` g ) ` x ) = { u } ) )
13	7, 12	rabeqbidv 2810	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | ( e ` x ) = { u } } = { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } )
14	13	fveq2d 5679	. . . . . . 7 |- ( e = (iEdg ` g ) -> (♯ ` { x e. dom e | ( e ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) )
15	11, 14	oveq12d 6076	. . . . . 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 4197	. . . 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 3191	. . 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 5675	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
20		vtxdgfval.v	. . . . . 6 |- V = (Vtx ` G )
21	19, 20	eqtr4di 2285	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
22		fveq2 5675	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
23	22	dmeqd 4963	. . . . . . . . 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 4962	. . . . . . . . . 10 |- dom I = dom (iEdg ` G )
27	24, 26	eqtri 2255	. . . . . . . . 9 |- A = dom (iEdg ` G )
28	23, 27	eqtr4di 2285	. . . . . . . 8 |- ( g = G -> dom (iEdg ` g ) = A )
29	22, 25	eqtr4di 2285	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = I )
30	29	fveq1d 5677	. . . . . . . . 9 |- ( g = G -> ( (iEdg ` g ) ` x ) = ( I ` x ) )
31	30	eleq2d 2304	. . . . . . . 8 |- ( g = G -> ( u e. ( (iEdg ` g ) ` x ) <-> u e. ( I ` x ) ) )
32	28, 31	rabeqbidv 2810	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } = { x e. A | u e. ( I ` x ) } )
33	32	fveq2d 5679	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) = (♯ ` { x e. A | u e. ( I ` x ) } ) )
34	30	eqeq1d 2243	. . . . . . . 8 |- ( g = G -> ( ( (iEdg ` g ) ` x ) = { u } <-> ( I ` x ) = { u } ) )
35	28, 34	rabeqbidv 2810	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } = { x e. A | ( I ` x ) = { u } } )
36	35	fveq2d 5679	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) = (♯ ` { x e. A | ( I ` x ) = { u } } ) )
37	33, 36	oveq12d 6076	. . . . 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 4197	. . . 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 2279	. 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 2827	. 2 |- ( G e. W -> G e. _V )
42		vtxex 16139	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
43	20, 42	eqeltrid 2321	. . 3 |- ( G e. W -> V e. _V )
44	43	mptexd 5918	. 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 5764	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 2205   {crab 2526   _Vcvv 2815   [_csb 3141   {csn 3694   |-> 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:  vtxdgfifival  16412  vtxdgop  16413  vtxdgfif  16414  vtxdeqd  16417