Theorem vtxdgfval 16734

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 16733	. 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 16464	. . . . 5 |- ( g e. _V -> (Vtx ` g ) e. _V )
3	2	elv 2825	. . . 4 |- (Vtx ` g ) e. _V
4		iedgex 16465	. . . . 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 4983	. . . . . . . . 9 |- ( e = (iEdg ` g ) -> dom e = dom (iEdg ` g ) )
8		fveq1 5697	. . . . . . . . . 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 5702	. . . . . . 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 5702	. . . . . . 7 |- ( e = (iEdg ` g ) -> (♯ ` { x e. dom e | ( e ` x ) = { u } } ) = (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) )
15	11, 14	oveq12d 6106	. . . . . 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 4214	. . . 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 5698	. . . . . 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 5698	. . . . . . . . . 10 |- ( g = G -> (iEdg ` g ) = (iEdg ` G ) )
23	22	dmeqd 4985	. . . . . . . . 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 4984	. . . . . . . . . 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 5700	. . . . . . . . 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 5702	. . . . . 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 5702	. . . . . 6 |- ( g = G -> (♯ ` { x e. dom (iEdg ` g ) | ( (iEdg ` g ) ` x ) = { u } } ) = (♯ ` { x e. A | ( I ` x ) = { u } } ) )
37	33, 36	oveq12d 6106	. . . . 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 4214	. . . 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 16464	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
43	20, 42	eqeltrid 2325	. . 3 |- ( G e. W -> V e. _V )
44	43	mptexd 5947	. 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 5790	1 |- ( G e. W -> (VtxDeg ` G ) = ( u e. V |-> ( (♯ ` { x e. A | u e. ( I ` x ) } ) +e (♯ ` { x e. A | ( I ` x ) = { u } } ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   _Vcvv 2821   [_csb 3147   {csn 3709   |-> cmpt 4193   dom cdm 4776   ` cfv 5380  (class class class)co 6088   +ecxad 10189  ♯chash 11236  Vtxcvtx 16458  iEdgciedg 16459  VtxDegcvtxdg 16732

This proof depends on 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 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-sub 8506  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-dec 9789  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-vtx 16460  df-iedg 16461  df-vtxdg 16733

This theorem is used by:  vtxdgfifival  16737  vtxdgop  16738  vtxdgfif  16739  vtxdeqd  16742