Theorem vdusgraval 25034

Description: The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)

Assertion

Ref	Expression
vdusgraval	|- ( ( V USGrph E /\ U e. V ) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) )

Distinct variable groups:   x, E   x, U   x, V

Proof of Theorem vdusgraval

Step	Hyp	Ref	Expression
1		usgranloop0 24507	. 2 |- ( ( V USGrph E /\ U e. V ) -> { x e. dom E | ( E ` x ) = { U } } = (/) )
2		usgrafun 24476	. . . . . . 7 |- ( V USGrph E -> Fun E )
3		usgrav 24465	. . . . . . 7 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
4		funfn 5623	. . . . . . . 8 |- ( Fun E <-> E Fn dom E )
5		simpl 457	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> V e. _V )
6	5	adantl 466	. . . . . . . . . 10 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> V e. _V )
7		simpl 457	. . . . . . . . . 10 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> E Fn dom E )
8		dmexg 6730	. . . . . . . . . . . 12 |- ( E e. _V -> dom E e. _V )
9	8	adantl 466	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> dom E e. _V )
10	9	adantl 466	. . . . . . . . . 10 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> dom E e. _V )
11	6, 7, 10	3jca 1176	. . . . . . . . 9 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> ( V e. _V /\ E Fn dom E /\ dom E e. _V ) )
12	11	ex 434	. . . . . . . 8 |- ( E Fn dom E -> ( ( V e. _V /\ E e. _V ) -> ( V e. _V /\ E Fn dom E /\ dom E e. _V ) ) )
13	4, 12	sylbi 195	. . . . . . 7 |- ( Fun E -> ( ( V e. _V /\ E e. _V ) -> ( V e. _V /\ E Fn dom E /\ dom E e. _V ) ) )
14	2, 3, 13	sylc 60	. . . . . 6 |- ( V USGrph E -> ( V e. _V /\ E Fn dom E /\ dom E e. _V ) )
15	14	anim1i 568	. . . . 5 |- ( ( V USGrph E /\ U e. V ) -> ( ( V e. _V /\ E Fn dom E /\ dom E e. _V ) /\ U e. V ) )
16		vdgrval 25023	. . . . 5 |- ( ( ( V e. _V /\ E Fn dom E /\ dom E e. _V ) /\ U e. V ) -> ( ( V VDeg E ) ` U ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) )
17		fveq2 5872	. . . . . . . . . . 11 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = ( # ` (/) ) )
18		hash0 12440	. . . . . . . . . . 11 |- ( # ` (/) ) = 0
19	17, 18	syl6eq 2514	. . . . . . . . . 10 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 )
20		oveq2 6304	. . . . . . . . . . . . 13 |- ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e 0 ) )
21	20	adantr 465	. . . . . . . . . . . 12 |- ( ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 /\ V USGrph E ) -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e 0 ) )
22	3	simprd 463	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> E e. _V )
23		rabexg 4606	. . . . . . . . . . . . . . 15 |- ( dom E e. _V -> { x e. dom E | U e. ( E ` x ) } e. _V )
24	22, 8, 23	3syl 20	. . . . . . . . . . . . . 14 |- ( V USGrph E -> { x e. dom E | U e. ( E ` x ) } e. _V )
25	24	adantl 466	. . . . . . . . . . . . 13 |- ( ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 /\ V USGrph E ) -> { x e. dom E | U e. ( E ` x ) } e. _V )
26		hashxrcl 12432	. . . . . . . . . . . . 13 |- ( { x e. dom E | U e. ( E ` x ) } e. _V -> ( # ` { x e. dom E | U e. ( E ` x ) } ) e. RR* )
27		xaddid1 11463	. . . . . . . . . . . . 13 |- ( ( # ` { x e. dom E | U e. ( E ` x ) } ) e. RR* -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e 0 ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) )
28	25, 26, 27	3syl 20	. . . . . . . . . . . 12 |- ( ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 /\ V USGrph E ) -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e 0 ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) )
29	21, 28	eqtrd 2498	. . . . . . . . . . 11 |- ( ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 /\ V USGrph E ) -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) )
30	29	ex 434	. . . . . . . . . 10 |- ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 -> ( V USGrph E -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
31	19, 30	syl 16	. . . . . . . . 9 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( V USGrph E -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
32	31	com12 31	. . . . . . . 8 |- ( V USGrph E -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
33	32	adantl 466	. . . . . . 7 |- ( ( ( ( V VDeg E ) ` U ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) /\ V USGrph E ) -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
34		eqeq1 2461	. . . . . . . 8 |- ( ( ( V VDeg E ) ` U ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) -> ( ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) <-> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
35	34	adantr 465	. . . . . . 7 |- ( ( ( ( V VDeg E ) ` U ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) /\ V USGrph E ) -> ( ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) <-> ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
36	33, 35	sylibrd 234	. . . . . 6 |- ( ( ( ( V VDeg E ) ` U ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) /\ V USGrph E ) -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
37	36	ex 434	. . . . 5 |- ( ( ( V VDeg E ) ` U ) = ( ( # ` { x e. dom E | U e. ( E ` x ) } ) +e ( # ` { x e. dom E | ( E ` x ) = { U } } ) ) -> ( V USGrph E -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) ) )
38	15, 16, 37	3syl 20	. . . 4 |- ( ( V USGrph E /\ U e. V ) -> ( V USGrph E -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) ) )
39	38	com12 31	. . 3 |- ( V USGrph E -> ( ( V USGrph E /\ U e. V ) -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) ) )
40	39	anabsi5 817	. 2 |- ( ( V USGrph E /\ U e. V ) -> ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) ) )
41	1, 40	mpd 15	1 |- ( ( V USGrph E /\ U e. V ) -> ( ( V VDeg E ) ` U ) = ( # ` { x e. dom E | U e. ( E ` x ) } ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   {crab 2811   _Vcvv 3109   (/)c0 3793   {csn 4032   class class class wbr 4456   dom cdm 5008   Fun wfun 5588   Fn wfn 5589   ` cfv 5594  (class class class)co 6296   0cc0 9509   RR*cxr 9644   +ecxad 11341   #chash 12408   USGrph cusg 24457   VDeg cvdg 25020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-xadd 11344  df-fz 11698  df-hash 12409  df-usgra 24460  df-vdgr 25021

This theorem is referenced by:  vdusgra0nedg  25035  hashnbgravd  25039  hashnbgravdg  25040  usgravd0nedg  25045  vdn0frgrav2  25150  vdgn0frgrav2  25151  vdn1frgrav2  25152  vdgn1frgrav2  25153