Theorem vtxlpfi 16214

Description: In a finite graph, the number of loops 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
vtxlpfi	|- ( ph -> { x e. A | ( I ` x ) = { U } } e. Fin )

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

Allowed substitution hint:   ph( x)

Proof of Theorem vtxlpfi

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 7099	. . . . . . . 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 2615	. . . . . 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 2298	. . . . . . 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 16023	. . . . . 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 16021	. . . . . . . . 9 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
22	21	ffnd 5490	. . . . . . . 8 |- ( G e. UPGraph -> I Fn dom I )
23	13	fneq2i 5432	. . . . . . . 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 16026	. . . . . 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	eqsndc 7138	. . . 4 |- ( ( ph /\ r e. A ) -> DECID ( I ` r ) = { U } )
30	29	ralrimiva 2606	. . 3 |- ( ph -> A. r e. A DECID ( I ` r ) = { U } )
31		fveqeq2 5657	. . . . 5 |- ( r = x -> ( ( I ` r ) = { U } <-> ( I ` x ) = { U } ) )
32	31	dcbid 846	. . . 4 |- ( r = x -> (DECID ( I ` r ) = { U } <-> DECID ( I ` x ) = { U } ) )
33	32	cbvralv 2768	. . 3 |- ( A. r e. A DECID ( I ` r ) = { U } <-> A. x e. A DECID ( I ` x ) = { U } )
34	30, 33	sylib 122	. 2 |- ( ph -> A. x e. A DECID ( I ` x ) = { U } )
35	1, 34	ssfirab 7172	1 |- ( ph -> { x e. A | ( I ` x ) = { U } } e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2202   A.wral 2511   {crab 2515   C_ wss 3201   ~Pcpw 3656   {csn 3673   class class class wbr 4093   dom cdm 4731   Fn wfn 5328   ` cfv 5333   1oc1o 6618   2oc2o 6619   ~~ cen 6950   Fincfn 6952  Vtxcvtx 15936  iEdgciedg 15937  UPGraphcupgr 16015

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953  df-fin 6955  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-upgren 16017

This theorem is referenced by:  vtxdgfifival  16215  vtxdgfif  16217  vtxdfifiun  16221  vtxd0nedgbfi  16223  vtxduspgrfvedgfi  16225