Theorem vtxdgfval 16443

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 16442	. 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 16173	. . . . 5 |- ( g e. _V -> (Vtx ` g ) e. _V )
3	2	elv 2825	. . . 4 |- (Vtx ` g ) e. _V
4		iedgex 16174	. . . . 5 |- ( g e. _V -> (iEdg ` g ) e. _V )
5	4	elv 2825	. . . 4 |- (iEdg ` g ) e. _V
6		simpl 109	. . . . 5 |- ( ( v = (Vtx ` g ) /\ e = (iEdg ` g ) ) -> v = (Vtx ` g ) )
7		dmeq 4976	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> dom e = dom (iEdg ` g ) )
8		fveq1 5689	. . . . . . . . . 10 |- ( e = (iEdg ` g ) -> ( e ` x ) = ( (iEdg ` g ) ` x ) )
9	8	eleq2d 2308	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( u e. ( e ` x ) <-> u e. ( (iEdg ` g ) ` x ) ) )
10	7, 9	rabeqbidv 2816	. . . . . . . 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 5694	. . . . . . 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 2247	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> ( ( e ` x ) = { u } <-> ( (iEdg ` g ) ` x ) = { u } ) )
13	7, 12	rabeqbidv 2816	. . . . . . . 8 |- ( e = (iEdg ` g ) -> { x e. dom e | ( e ` x ) = { u } } = { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } )
14	13	fveq2d 5694	. . . . . . 7 |- ( e = (iEdg ` g ) -> (♯ ` { x e. dom e | ( e ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) )
15	11, 14	oveq12d 6093	. . . . . 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 4208	. . . 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 3197	. . 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 5690	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
20		vtxdgfval.v	. . . . . 6 |- V = (Vtx ` G )
21	19, 20	eqtr4di 2289	. . . . 5 |- ( g = G -> (Vtx ` g ) = V )
22		fveq2 5690	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
23	22	dmeqd 4978	. . . . . . . . 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 4977	. . . . . . . . . 10 |- dom I = dom (iEdg ` G )
27	24, 26	eqtri 2259	. . . . . . . . 9 |- A = dom (iEdg ` G )
28	23, 27	eqtr4di 2289	. . . . . . . 8 |- ( g = G -> dom (iEdg ` g ) = A )
29	22, 25	eqtr4di 2289	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = I )
30	29	fveq1d 5692	. . . . . . . . 9 |- ( g = G -> ( (iEdg ` g ) ` x ) = ( I ` x ) )
31	30	eleq2d 2308	. . . . . . . 8 |- ( g = G -> ( u e. ( (iEdg ` g ) ` x ) <-> u e. ( I ` x ) ) )
32	28, 31	rabeqbidv 2816	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } = { x e. A | u e. ( I ` x ) } )
33	32	fveq2d 5694	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | u e. ( (iEdg ` g ) ` x ) } ) = (♯ ` { x e. A | u e. ( I ` x ) } ) )
34	30	eqeq1d 2247	. . . . . . . 8 |- ( g = G -> ( ( (iEdg ` g ) ` x ) = { u } <-> ( I ` x ) = { u } ) )
35	28, 34	rabeqbidv 2816	. . . . . . 7 |- ( g = G -> { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } = { x e. A | ( I ` x ) = { u } } )
36	35	fveq2d 5694	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) = (♯ ` { x e. A | ( I ` x ) = { u } } ) )
37	33, 36	oveq12d 6093	. . . . 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 4208	. . . 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 2283	. 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 2833	. 2 |- ( G e. W -> G e. _V )
42		vtxex 16173	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
43	20, 42	eqeltrid 2325	. . 3 |- ( G e. W -> V e. _V )
44	43	mptexd 5935	. 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 5781	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 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   [_csb 3147   {csn 3705   |-> cmpt 4187   dom cdm 4769   ` cfv 5372  (class class class)co 6075   +ecxad 10151  ♯chash 11192  Vtxcvtx 16167  iEdgciedg 16168  VtxDegcvtxdg 16441

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-vtxdg 16442

This theorem is referenced by:  vtxdgfifival  16446  vtxdgop  16447  vtxdgfif  16448  vtxdeqd  16451