Theorem vtxduspgrfvedgfi 16422

Description: The value of the vertex degree function for a simple pseudograph. (Contributed by AV, 12-Dec-2020.) (Proof shortened by AV, 5-May-2021.)

Hypotheses

Ref	Expression
vtxdushgrfvedg.v	|- V = (Vtx ` G )
vtxdushgrfvedg.e	|- E = (Edg ` G )
vtxduspgrfvedgfi.fi	|- ( ph -> dom (iEdg ` G ) e. Fin )
vtxduspgrfvedgfi.v	|- ( ph -> V e. Fin )
vtxduspgrfvedgfi.u	|- ( ph -> U e. V )
vtxduspgrfvedgfi.g	|- ( ph -> G e. USPGraph )
vtxdushgrfvedg.d	|- D = (VtxDeg ` G )

Assertion

Ref	Expression
vtxduspgrfvedgfi	|- ( ph -> ( D ` U ) = ( (♯ ` { e e. E | U e. e } ) + (♯ ` { e e. E | e = { U } } ) ) )

Distinct variable groups:   e, E   e, G   U, e   e, V

Allowed substitution hints:   ph( e)   D( e)

Proof of Theorem vtxduspgrfvedgfi

Dummy variables c i x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdushgrfvedg.d	. . . 4 |- D = (VtxDeg ` G )
2	1	fveq1i 5676	. . 3 |- ( D ` U ) = ( (VtxDeg ` G ) ` U )
3	2	a1i 9	. 2 |- ( ph -> ( D ` U ) = ( (VtxDeg ` G ) ` U ) )
4		vtxdushgrfvedg.v	. . 3 |- V = (Vtx ` G )
5		eqid 2234	. . 3 |- (iEdg ` G ) = (iEdg ` G )
6		eqid 2234	. . 3 |- dom (iEdg ` G ) = dom (iEdg ` G )
7		vtxduspgrfvedgfi.fi	. . 3 |- ( ph -> dom (iEdg ` G ) e. Fin )
8		vtxduspgrfvedgfi.v	. . 3 |- ( ph -> V e. Fin )
9		vtxduspgrfvedgfi.u	. . 3 |- ( ph -> U e. V )
10		vtxduspgrfvedgfi.g	. . . 4 |- ( ph -> G e. USPGraph )
11		uspgrupgr 16302	. . . 4 |- ( G e. USPGraph -> G e. UPGraph )
12	10, 11	syl 14	. . 3 |- ( ph -> G e. UPGraph )
13	4, 5, 6, 7, 8, 9, 12	vtxdgfifival 16412	. 2 |- ( ph -> ( (VtxDeg ` G ) ` U ) = ( (♯ ` { i e. dom (iEdg ` G ) | U e. ( (iEdg ` G ) ` i ) } ) + (♯ ` { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } ) ) )
14		vtxdushgrfvedg.e	. . . 4 |- E = (Edg ` G )
15	4, 14, 7, 8, 9, 10	vtxduspgrfvedgfilem 16421	. . 3 |- ( ph -> (♯ ` { i e. dom (iEdg ` G ) | U e. ( (iEdg ` G ) ` i ) } ) = (♯ ` { e e. E | U e. e } ) )
16	4, 5, 6, 7, 8, 9, 12	vtxlpfi 16411	. . . 4 |- ( ph -> { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } e. Fin )
17		uspgrushgr 16301	. . . . . 6 |- ( G e. USPGraph -> G e. USHGraph )
18	10, 17	syl 14	. . . . 5 |- ( ph -> G e. USHGraph )
19		eqid 2234	. . . . . 6 |- { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } = { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } }
20		eqeq1 2241	. . . . . . 7 |- ( e = c -> ( e = { U } <-> c = { U } ) )
21	20	cbvrabv 2814	. . . . . 6 |- { e e. E | e = { U } } = { c e. E | c = { U } }
22		eqid 2234	. . . . . 6 |- ( x e. { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } |-> ( (iEdg ` G ) ` x ) ) = ( x e. { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } |-> ( (iEdg ` G ) ` x ) )
23	14, 5, 19, 21, 22	ushgredgedgloop 16349	. . . . 5 |- ( ( G e. USHGraph /\ U e. V ) -> ( x e. { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } |-> ( (iEdg ` G ) ` x ) ) : { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } -1-1-onto-> { e e. E | e = { U } } )
24	18, 9, 23	syl2anc 411	. . . 4 |- ( ph -> ( x e. { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } |-> ( (iEdg ` G ) ` x ) ) : { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } -1-1-onto-> { e e. E | e = { U } } )
25	16, 24	fihasheqf1od 11177	. . 3 |- ( ph -> (♯ ` { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } ) = (♯ ` { e e. E | e = { U } } ) )
26	15, 25	oveq12d 6076	. 2 |- ( ph -> ( (♯ ` { i e. dom (iEdg ` G ) | U e. ( (iEdg ` G ) ` i ) } ) + (♯ ` { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } ) ) = ( (♯ ` { e e. E | U e. e } ) + (♯ ` { e e. E | e = { U } } ) ) )
27	3, 13, 26	3eqtrd 2271	1 |- ( ph -> ( D ` U ) = ( (♯ ` { e e. E | U e. e } ) + (♯ ` { e e. E | e = { U } } ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {crab 2526   {csn 3694   |-> cmpt 4176   dom cdm 4754   -1-1-onto->wf1o 5356   ` cfv 5357  (class class class)co 6058   Fincfn 6988   + caddc 8146  ♯chash 11163  Vtxcvtx 16133  iEdgciedg 16134  Edgcedg 16178  USHGraphcushgr 16189  UPGraphcupgr 16212  USPGraphcuspgr 16274  VtxDegcvtxdg 16407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-xadd 10125  df-ihash 11164  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-edg 16179  df-uhgrm 16190  df-ushgrm 16191  df-upgren 16214  df-uspgren 16276  df-vtxdg 16408

This theorem is referenced by:  1loopgrvd2fi  16426