Theorem vtxdgval 26364

Description: The degree of a vertex. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.) (Revised by AV, 10-Dec-2020.) (Revised by AV, 22-Mar-2021.)

Hypotheses

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

Assertion

Ref	Expression
vtxdgval	|- ( U e. V -> ( (VtxDeg ` G ) ` U ) = ( ( # ` { x e. A | U e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { U } } ) ) )

Distinct variable groups:   x, A   x, G   x, U

Allowed substitution hints:   I( x)   V( x)

Proof of Theorem vtxdgval

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdgval.v	. . . . 5 |- V = (Vtx ` G )
2	1	1vgrex 25882	. . . 4 |- ( U e. V -> G e. _V )
3		vtxdgval.i	. . . . 5 |- I = (iEdg ` G )
4		vtxdgval.a	. . . . 5 |- A = dom I
5	1, 3, 4	vtxdgfval 26363	. . . 4 |- ( G e. _V -> (VtxDeg ` G ) = ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) )
6	2, 5	syl 17	. . 3 |- ( U e. V -> (VtxDeg ` G ) = ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) )
7	6	fveq1d 6193	. 2 |- ( U e. V -> ( (VtxDeg ` G ) ` U ) = ( ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) ` U ) )
8		eleq1 2689	. . . . . 6 |- ( u = U -> ( u e. ( I ` x ) <-> U e. ( I ` x ) ) )
9	8	rabbidv 3189	. . . . 5 |- ( u = U -> { x e. A | u e. ( I ` x ) } = { x e. A | U e. ( I ` x ) } )
10	9	fveq2d 6195	. . . 4 |- ( u = U -> ( # ` { x e. A | u e. ( I ` x ) } ) = ( # ` { x e. A | U e. ( I ` x ) } ) )
11		sneq 4187	. . . . . . 7 |- ( u = U -> { u } = { U } )
12	11	eqeq2d 2632	. . . . . 6 |- ( u = U -> ( ( I ` x ) = { u } <-> ( I ` x ) = { U } ) )
13	12	rabbidv 3189	. . . . 5 |- ( u = U -> { x e. A | ( I ` x ) = { u } } = { x e. A | ( I ` x ) = { U } } )
14	13	fveq2d 6195	. . . 4 |- ( u = U -> ( # ` { x e. A | ( I ` x ) = { u } } ) = ( # ` { x e. A | ( I ` x ) = { U } } ) )
15	10, 14	oveq12d 6668	. . 3 |- ( u = U -> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) = ( ( # ` { x e. A | U e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { U } } ) ) )
16		eqid 2622	. . 3 |- ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) = ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) )
17		ovex 6678	. . 3 |- ( ( # ` { x e. A | U e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { U } } ) ) e. _V
18	15, 16, 17	fvmpt 6282	. 2 |- ( U e. V -> ( ( u e. V |-> ( ( # ` { x e. A | u e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { u } } ) ) ) ` U ) = ( ( # ` { x e. A | U e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { U } } ) ) )
19	7, 18	eqtrd 2656	1 |- ( U e. V -> ( (VtxDeg ` G ) ` U ) = ( ( # ` { x e. A | U e. ( I ` x ) } ) +e ( # ` { x e. A | ( I ` x ) = { U } } ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   {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-8 1992  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-pow 4843  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:  vtxdgfival  26365  vtxdun  26377  vtxdlfgrval  26381  vtxd0nedgb  26384  vtxdushgrfvedg  26386  vtxdginducedm1  26439