Theorem vtxduspgrfvedgfi 16456

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 5691	. . 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 2238	. . 3 |- (iEdg ` G ) = (iEdg ` G )
6		eqid 2238	. . 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 16336	. . . 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 16446	. 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 16455	. . 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 16445	. . . 4 |- ( ph -> { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } e. Fin )
17		uspgrushgr 16335	. . . . . 6 |- ( G e. USPGraph -> G e. USHGraph )
18	10, 17	syl 14	. . . . 5 |- ( ph -> G e. USHGraph )
19		eqid 2238	. . . . . 6 |- { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } = { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } }
20		eqeq1 2245	. . . . . . 7 |- ( e = c -> ( e = { U } <-> c = { U } ) )
21	20	cbvrabv 2820	. . . . . 6 |- { e e. E | e = { U } } = { c e. E | c = { U } }
22		eqid 2238	. . . . . 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 16383	. . . . 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 415	. . . 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 11206	. . 3 |- ( ph -> (♯ ` { i e. dom (iEdg ` G ) | ( (iEdg ` G ) ` i ) = { U } } ) = (♯ ` { e e. E | e = { U } } ) )
26	15, 25	oveq12d 6093	. 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 2275	1 |- ( ph -> ( D ` U ) = ( (♯ ` { e e. E | U e. e } ) + (♯ ` { e e. E | e = { U } } ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {crab 2532   {csn 3705   |-> cmpt 4187   dom cdm 4769   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075   Fincfn 7012   + caddc 8172  ♯chash 11192  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  USHGraphcushgr 16223  UPGraphcupgr 16246  USPGraphcuspgr 16308  VtxDegcvtxdg 16441

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-2o 6678  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-z 9624  df-dec 9757  df-uz 9901  df-xadd 10154  df-ihash 11193  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-ushgrm 16225  df-upgren 16248  df-uspgren 16310  df-vtxdg 16442

This theorem is referenced by:  1loopgrvd2fi  16460