Theorem vdusgraval 23724

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 23446	. 2 |- ( ( V USGrph E /\ U e. V ) -> { x e. dom E | ( E ` x ) = { U } } = (/) )
2		usgrafun 23424	. . . . . . 7 |- ( V USGrph E -> Fun E )
3		usgrav 23417	. . . . . . 7 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
4		funfn 5550	. . . . . . . 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 6614	. . . . . . . . . . . 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 1168	. . . . . . . . 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 23713	. . . . 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 5794	. . . . . . . . . . 11 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = ( # ` (/) ) )
18		hash0 12247	. . . . . . . . . . 11 |- ( # ` (/) ) = 0
19	17, 18	syl6eq 2509	. . . . . . . . . 10 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 )
20		oveq2 6203	. . . . . . . . . . . . 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 4545	. . . . . . . . . . . . . . 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 12239	. . . . . . . . . . . . 13 |- ( { x e. dom E | U e. ( E ` x ) } e. _V -> ( # ` { x e. dom E | U e. ( E ` x ) } ) e. RR* )
27		xaddid1 11315	. . . . . . . . . . . . 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 2493	. . . . . . . . . . 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 2456	. . . . . . . 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 813	. 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 965   = wceq 1370   e. wcel 1758   {crab 2800   _Vcvv 3072   (/)c0 3740   {csn 3980   class class class wbr 4395   dom cdm 4943   Fun wfun 5515   Fn wfn 5516   ` cfv 5521  (class class class)co 6195   0cc0 9388   RR*cxr 9523   +ecxad 11193   #chash 12215   USGrph cusg 23411   VDeg cvdg 23710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-n0 10686  df-z 10753  df-uz 10968  df-xadd 11196  df-fz 11550  df-hash 12216  df-usgra 23413  df-vdgr 23711

This theorem is referenced by:  vdusgra0nedg  23725  hashnbgravd  23727  hashnbgravdg  23728  usgravd0nedg  23729  vdn0frgrav2  30759  vdgn0frgrav2  30760  vdn1frgrav2  30761  vdgn1frgrav2  30762