Theorem vtxedgfi 16048

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 529	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> p e. V )
5		simprr 531	. . . . . . . 8 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> q e. V )
6		fidceq 7039	. . . . . . . 8 |- ( ( V e. Fin /\ p e. V /\ q e. V ) -> DECID p = q )
7	3, 4, 5, 6	syl3anc 1271	. . . . . . 7 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> DECID p = q )
8	7	ralrimivva 2612	. . . . . 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 2296	. . . . . . 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 15914	. . . . . 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 15912	. . . . . . . . 9 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
22	21	ffnd 5474	. . . . . . . 8 |- ( G e. UPGraph -> I Fn dom I )
23	13	fneq2i 5416	. . . . . . . 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 15917	. . . . . 6 |- ( ( G e. UPGraph /\ I Fn A /\ r e. A ) -> ( I ` r ) e. Fin )
28	20, 25, 26, 27	syl3anc 1271	. . . . 5 |- ( ( ph /\ r e. A ) -> ( I ` r ) e. Fin )
29	9, 11, 19, 28	elssdc 7075	. . . 4 |- ( ( ph /\ r e. A ) -> DECID U e. ( I ` r ) )
30	29	ralrimiva 2603	. . 3 |- ( ph -> A. r e. A DECID U e. ( I ` r ) )
31		fveq2 5629	. . . . . 6 |- ( r = x -> ( I ` r ) = ( I ` x ) )
32	31	eleq2d 2299	. . . . 5 |- ( r = x -> ( U e. ( I ` r ) <-> U e. ( I ` x ) ) )
33	32	dcbid 843	. . . 4 |- ( r = x -> (DECID U e. ( I ` r ) <-> DECID U e. ( I ` x ) ) )
34	33	cbvralv 2765	. . 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 7109	1 |- ( ph -> { x e. A | U e. ( I ` x ) } e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   A.wral 2508   {crab 2512   C_ wss 3197   ~Pcpw 3649   class class class wbr 4083   dom cdm 4719   Fn wfn 5313   ` cfv 5318   1oc1o 6561   2oc2o 6562   ~~ cen 6893   Fincfn 6895  Vtxcvtx 15828  iEdgciedg 15829  UPGraphcupgr 15906

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-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

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-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-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-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-fin 6898  df-sub 8330  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-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831  df-upgren 15908

This theorem is referenced by:  vtxdgfifival  16050  vtxdgfif  16052  vtxdfifiun  16056  vtxdumgrfival  16057