Theorem vtxdgfval 26363

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 26362	. . 3 |- 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	1	a1i 11	. 2 |- ( G e. W -> 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 } } ) ) ) ) )
3		fvex 6201	. . . 4 |- (Vtx ` g ) e. _V
4		fvex 6201	. . . 4 |- (iEdg ` g ) e. _V
5		simpl 473	. . . . 5 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> v = (Vtx ` g ) )
6		dmeq 5324	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> dom e = dom (iEdg ` g ) )
7		fveq1 6190	. . . . . . . . . 10 |- ( e = (iEdg ` g ) -> ( e ` x ) = ( (iEdg ` g ) ` x ) )
8	7	eleq2d 2687	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( u e. ( e ` x ) <-> u e. ( (iEdg ` g ) ` x ) ) )
9	6, 8	rabeqbidv 3195	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | u e. ( e ` x ) } = { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } )
10	9	fveq2d 6195	. . . . . . 7 |- ( e = (iEdg ` g ) -> ( # ` { x e. dom e | u e. ( e ` x ) } ) = ( # ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) )
11	7	eqeq1d 2624	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( ( e ` x ) = { u } <-> ( (iEdg ` g ) ` x ) = { u } ) )
12	6, 11	rabeqbidv 3195	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | ( e ` x ) = { u } } = { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } )
13	12	fveq2d 6195	. . . . . . 7 |- ( e = (iEdg ` g ) -> ( # ` { x e. dom e | ( e ` x ) = { u } } ) = ( # ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) )
14	10, 13	oveq12d 6668	. . . . . 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 } } ) ) )
15	14	adantl 482	. . . . 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 } } ) ) )
16	5, 15	mpteq12dv 4733	. . . 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 } } ) ) ) )
17	3, 4, 16	csbie2 3563	. . 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 } } ) ) )
18		fveq2 6191	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
19		vtxdgfval.v	. . . . . 6 |- V = (Vtx ` G )
20	18, 19	syl6eqr 2674	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
21		fveq2 6191	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
22	21	dmeqd 5326	. . . . . . . . 9 |- ( g = G -> dom (iEdg ` g ) = dom (iEdg ` G ) )
23		vtxdgfval.a	. . . . . . . . . 10 |- A = dom I
24		vtxdgfval.i	. . . . . . . . . . 11 |- I = (iEdg ` G )
25	24	dmeqi 5325	. . . . . . . . . 10 |- dom I = dom (iEdg ` G )
26	23, 25	eqtri 2644	. . . . . . . . 9 |- A = dom (iEdg ` G )
27	22, 26	syl6eqr 2674	. . . . . . . 8 |- ( g = G -> dom (iEdg ` g ) = A )
28	21, 24	syl6eqr 2674	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = I )
29	28	fveq1d 6193	. . . . . . . . 9 |- ( g = G -> ( (iEdg ` g ) ` x ) = ( I ` x ) )
30	29	eleq2d 2687	. . . . . . . 8 |- ( g = G -> ( u e. ( (iEdg ` g ) ` x ) <-> u e. ( I ` x ) ) )
31	27, 30	rabeqbidv 3195	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } = { x e. A | u e. ( I ` x ) } )
32	31	fveq2d 6195	. . . . . 6 |- ( g = G -> ( # ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) = ( # ` { x e. A | u e. ( I ` x ) } ) )
33	29	eqeq1d 2624	. . . . . . . 8 |- ( g = G -> ( ( (iEdg ` g ) ` x ) = { u } <-> ( I ` x ) = { u } ) )
34	27, 33	rabeqbidv 3195	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } = { x e. A | ( I ` x ) = { u } } )
35	34	fveq2d 6195	. . . . . 6 |- ( g = G -> ( # ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) = ( # ` { x e. A | ( I ` x ) = { u } } ) )
36	32, 35	oveq12d 6668	. . . . 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 } } ) ) )
37	20, 36	mpteq12dv 4733	. . . 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 } } ) ) ) )
38	37	adantl 482	. . 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 } } ) ) ) )
39	17, 38	syl5eq 2668	. 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 } } ) ) ) )
40		elex 3212	. 2 |- ( G e. W -> G e. _V )
41		fvex 6201	. . . 4 |- (Vtx ` G ) e. _V
42	19, 41	eqeltri 2697	. . 3 |- V e. _V
43		mptexg 6484	. . 3 |- ( V e. _V -> ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) e. _V )
44	42, 43	mp1i 13	. 2 |- ( G e. W -> ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) e. _V )
45	2, 39, 40, 44	fvmptd 6288	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 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   [_csb 3533   {csn 4177   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   +ecxad 11944   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  VtxDegcvtxdg 26361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-vtxdg 26362

This theorem is referenced by:  vtxdgval  26364  vtxdgop  26366  vtxdgf  26367  vtxdeqd  26373