Theorem vdusgraval 25628

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 25100	. 2 |- ( ( V USGrph E /\ U e. V ) -> { x e. dom E | ( E ` x ) = { U } } = (/) )
2		usgrafun 25069	. . . . . . 7 |- ( V USGrph E -> Fun E )
3		usgrav 25058	. . . . . . 7 |- ( V USGrph E -> ( V e. _V /\ E e. _V ) )
4		funfn 5610	. . . . . . . 8 |- ( Fun E <-> E Fn dom E )
5		simpl 459	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> V e. _V )
6	5	adantl 468	. . . . . . . . . 10 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> V e. _V )
7		simpl 459	. . . . . . . . . 10 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> E Fn dom E )
8		dmexg 6721	. . . . . . . . . . . 12 |- ( E e. _V -> dom E e. _V )
9	8	adantl 468	. . . . . . . . . . 11 |- ( ( V e. _V /\ E e. _V ) -> dom E e. _V )
10	9	adantl 468	. . . . . . . . . 10 |- ( ( E Fn dom E /\ ( V e. _V /\ E e. _V ) ) -> dom E e. _V )
11	6, 7, 10	3jca 1187	. . . . . . . . 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 436	. . . . . . . 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 199	. . . . . . 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 62	. . . . . 6 |- ( V USGrph E -> ( V e. _V /\ E Fn dom E /\ dom E e. _V ) )
15	14	anim1i 571	. . . . 5 |- ( ( V USGrph E /\ U e. V ) -> ( ( V e. _V /\ E Fn dom E /\ dom E e. _V ) /\ U e. V ) )
16		vdgrval 25617	. . . . 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 5863	. . . . . . . . . . 11 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = ( # ` (/) ) )
18		hash0 12545	. . . . . . . . . . 11 |- ( # ` (/) ) = 0
19	17, 18	syl6eq 2500	. . . . . . . . . 10 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 )
20		oveq2 6296	. . . . . . . . . . . . 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 467	. . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 15 |- ( V USGrph E -> E e. _V )
23		rabexg 4552	. . . . . . . . . . . . . . 15 |- ( dom E e. _V -> { x e. dom E | U e. ( E ` x ) } e. _V )
24	22, 8, 23	3syl 18	. . . . . . . . . . . . . 14 |- ( V USGrph E -> { x e. dom E | U e. ( E ` x ) } e. _V )
25	24	adantl 468	. . . . . . . . . . . . 13 |- ( ( ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 /\ V USGrph E ) -> { x e. dom E | U e. ( E ` x ) } e. _V )
26		hashxrcl 12536	. . . . . . . . . . . . 13 |- ( { x e. dom E | U e. ( E ` x ) } e. _V -> ( # ` { x e. dom E | U e. ( E ` x ) } ) e. RR* )
27		xaddid1 11529	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 2484	. . . . . . . . . . 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 436	. . . . . . . . . 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 17	. . . . . . . . 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 32	. . . . . . . 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 468	. . . . . . 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 2454	. . . . . . . 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 467	. . . . . . 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 238	. . . . . 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 436	. . . . 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 18	. . . 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 32	. . 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 825	. 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 188   /\ wa 371   /\ w3a 984   = wceq 1443   e. wcel 1886   {crab 2740   _Vcvv 3044   (/)c0 3730   {csn 3967   class class class wbr 4401   dom cdm 4833   Fun wfun 5575   Fn wfn 5576   ` cfv 5581  (class class class)co 6288   0cc0 9536   RR*cxr 9671   +ecxad 11404   #chash 12512   USGrph cusg 25050   VDeg cvdg 25614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-xadd 11407  df-fz 11782  df-hash 12513  df-usgra 25053  df-vdgr 25615

This theorem is referenced by:  vdusgra0nedg  25629  hashnbgravd  25633  hashnbgravdg  25634  usgravd0nedg  25639  vdn0frgrav2  25744  vdgn0frgrav2  25745  vdn1frgrav2  25746  vdgn1frgrav2  25747