Theorem vdusgraval 24721

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 24194	. 2 |- ( ( V USGrph E /\ U e. V ) -> { x e. dom E | ( E ` x ) = { U } } = (/) )
2		usgrafun 24163	. . . . . . 7 |- ( V USGrph E -> Fun E )
3		usgrav 24152	. . . . . . 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 6726	. . . . . . . . . . . 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 24710	. . . . 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 12417	. . . . . . . . . . 11 |- ( # ` (/) ) = 0
19	17, 18	syl6eq 2524	. . . . . . . . . 10 |- ( { x e. dom E | ( E ` x ) = { U } } = (/) -> ( # ` { x e. dom E | ( E ` x ) = { U } } ) = 0 )
20		oveq2 6303	. . . . . . . . . . . . 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 4603	. . . . . . . . . . . . . . 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 12409	. . . . . . . . . . . . 13 |- ( { x e. dom E | U e. ( E ` x ) } e. _V -> ( # ` { x e. dom E | U e. ( E ` x ) } ) e. RR* )
27		xaddid1 11450	. . . . . . . . . . . . 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 2508	. . . . . . . . . . 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 2471	. . . . . . . 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 815	. 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 1379   e. wcel 1767   {crab 2821   _Vcvv 3118   (/)c0 3790   {csn 4033   class class class wbr 4453   dom cdm 5005   Fun wfun 5588   Fn wfn 5589   ` cfv 5594  (class class class)co 6295   0cc0 9504   RR*cxr 9639   +ecxad 11328   #chash 12385   USGrph cusg 24144   VDeg cvdg 24707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-xadd 11331  df-fz 11685  df-hash 12386  df-usgra 24147  df-vdgr 24708

This theorem is referenced by:  vdusgra0nedg  24722  hashnbgravd  24726  hashnbgravdg  24727  usgravd0nedg  24732  vdn0frgrav2  24838  vdgn0frgrav2  24839  vdn1frgrav2  24840  vdgn1frgrav2  24841