Theorem vtxedgfi 16410

Description: In a finite graph, the number of edges from a given vertex is finite. (Contributed by Jim Kingdon, 16-Feb-2026.)

Hypotheses

Ref	Expression
vtxdgval.v	|- V = (Vtx ` G )
vtxdgval.i	|- I = (iEdg ` G )
vtxdgval.a	|- A = dom I
vtxdgfifival.a	|- ( ph -> A e. Fin )
vtxdgfifival.v	|- ( ph -> V e. Fin )
vtxdgfifival.u	|- ( ph -> U e. V )
vtxdgfifival.g	|- ( ph -> G e. UPGraph )

Assertion

Ref	Expression
vtxedgfi	|- ( ph -> { x e. A | U e. ( I ` x ) } e. Fin )

Distinct variable groups:   x, A   x, G   x, I   x, U   x, V

Allowed substitution hint:   ph( x)

Proof of Theorem vtxedgfi

Dummy variables r p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtxdgfifival.a	. 2 |- ( ph -> A e. Fin )
2		vtxdgfifival.v	. . . . . . . . 9 |- ( ph -> V e. Fin )
3	2	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> V e. Fin )
4		simprl 531	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> p e. V )
5		simprr 533	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> q e. V )
6		fidceq 7137	. . . . . . . 8 |- ( ( V e. Fin /\ p e. V /\ q e. V ) -> DECID p = q )
7	3, 4, 5, 6	syl3anc 1274	. . . . . . 7 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> DECID p = q )
8	7	ralrimivva 2626	. . . . . 6 |- ( ph -> A. p e. V A. q e. V DECID p = q )
9	8	adantr 276	. . . . 5 |- ( ( ph /\ r e. A ) -> A. p e. V A. q e. V DECID p = q )
10		vtxdgfifival.u	. . . . . 6 |- ( ph -> U e. V )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ r e. A ) -> U e. V )
12		vtxdgfifival.g	. . . . . 6 |- ( ph -> G e. UPGraph )
13		vtxdgval.a	. . . . . . . 8 |- A = dom I
14	13	eleq2i 2301	. . . . . . 7 |- ( r e. A <-> r e. dom I )
15	14	biimpi 120	. . . . . 6 |- ( r e. A -> r e. dom I )
16		vtxdgval.v	. . . . . . 7 |- V = (Vtx ` G )
17		vtxdgval.i	. . . . . . 7 |- I = (iEdg ` G )
18	16, 17	upgrss 16220	. . . . . 6 |- ( ( G e. UPGraph /\ r e. dom I ) -> ( I ` r ) C_ V )
19	12, 15, 18	syl2an 289	. . . . 5 |- ( ( ph /\ r e. A ) -> ( I ` r ) C_ V )
20	12	adantr 276	. . . . . 6 |- ( ( ph /\ r e. A ) -> G e. UPGraph )
21	16, 17	upgrfen 16218	. . . . . . . . 9 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
22	21	ffnd 5514	. . . . . . . 8 |- ( G e. UPGraph -> I Fn dom I )
23	13	fneq2i 5456	. . . . . . . 8 |- ( I Fn A <-> I Fn dom I )
24	22, 23	sylibr 134	. . . . . . 7 |- ( G e. UPGraph -> I Fn A )
25	20, 24	syl 14	. . . . . 6 |- ( ( ph /\ r e. A ) -> I Fn A )
26		simpr 110	. . . . . 6 |- ( ( ph /\ r e. A ) -> r e. A )
27	16, 17	upgrfi 16223	. . . . . 6 |- ( ( G e. UPGraph /\ I Fn A /\ r e. A ) -> ( I ` r ) e. Fin )
28	20, 25, 26, 27	syl3anc 1274	. . . . 5 |- ( ( ph /\ r e. A ) -> ( I ` r ) e. Fin )
29	9, 11, 19, 28	elssdc 7175	. . . 4 |- ( ( ph /\ r e. A ) -> DECID U e. ( I ` r ) )
30	29	ralrimiva 2617	. . 3 |- ( ph -> A. r e. A DECID U e. ( I ` r ) )
31		fveq2 5675	. . . . . 6 |- ( r = x -> ( I ` r ) = ( I ` x ) )
32	31	eleq2d 2304	. . . . 5 |- ( r = x -> ( U e. ( I ` r ) <-> U e. ( I ` x ) ) )
33	32	dcbid 846	. . . 4 |- ( r = x -> (DECID U e. ( I ` r ) <-> DECID U e. ( I ` x ) ) )
34	33	cbvralv 2780	. . 3 |- ( A. r e. A DECID U e. ( I ` r ) <-> A. x e. A DECID U e. ( I ` x ) )
35	30, 34	sylib 122	. 2 |- ( ph -> A. x e. A DECID U e. ( I ` x ) )
36	1, 35	ssfirab 7210	1 |- ( ph -> { x e. A | U e. ( I ` x ) } e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   C_ wss 3214   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   Fn wfn 5352   ` cfv 5357   1oc1o 6653   2oc2o 6654   ~~ cen 6986   Fincfn 6988  Vtxcvtx 16133  iEdgciedg 16134  UPGraphcupgr 16212

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-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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-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-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-1o 6660  df-2o 6661  df-er 6780  df-en 6989  df-fin 6991  df-sub 8462  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-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136  df-upgren 16214

This theorem is referenced by:  vtxdgfifival  16412  vtxdgfif  16414  vtxdfifiun  16418  vtxdumgrfival  16419  vtxd0nedgbfi  16420  vtxduspgrfvedgfilem  16421