Theorem vtxduspgrfvedgfi 16060

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 5630	. . 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 2229	. . 3 |- (iEdg ` G ) = (iEdg ` G )
6		eqid 2229	. . 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 15994	. . . 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 16050	. 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 16059	. . 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 16049	. . . 4 |- ( ph -> { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } e. Fin )
17		uspgrushgr 15993	. . . . . 6 |- ( G e. USPGraph -> G e. USHGraph )
18	10, 17	syl 14	. . . . 5 |- ( ph -> G e. USHGraph )
19		eqid 2229	. . . . . 6 |- { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } = { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } }
20		eqeq1 2236	. . . . . . 7 |- ( e = c -> ( e = { U } <-> c = { U } ) )
21	20	cbvrabv 2798	. . . . . 6 |- { e e. E | e = { U } } = { c e. E | c = { U } }
22		eqid 2229	. . . . . 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 16041	. . . . 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 11023	. . 3 |- ( ph -> (♯ ` { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } ) = (♯ ` { e e. E | e = { U } } ) )
26	15, 25	oveq12d 6025	. 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 2266	1 |- ( ph -> ( D ` U ) = ( (♯ ` { e e. E | U e. e } ) + (♯ ` { e e. E | e = { U } } ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {crab 2512   {csn 3666   |-> cmpt 4145   dom cdm 4719   -1-1-onto->wf1o 5317   ` cfv 5318  (class class class)co 6007   Fincfn 6895   + caddc 8013  ♯chash 11009  Vtxcvtx 15828  iEdgciedg 15829  Edgcedg 15873  USHGraphcushgr 15883  UPGraphcupgr 15906  USPGraphcuspgr 15966  VtxDegcvtxdg 16045

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-z 9458  df-dec 9590  df-uz 9734  df-xadd 9981  df-ihash 11010  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-edg 15874  df-uhgrm 15884  df-ushgrm 15885  df-upgren 15908  df-uspgren 15968  df-vtxdg 16046

This theorem is referenced by: (None)