Theorem vtxlpfi 16285

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 7124	. . . . . . . 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 2624	. . . . . 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 2299	. . . . . . 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 16094	. . . . . 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 16092	. . . . . . . . 9 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
22	21	ffnd 5509	. . . . . . . 8 |- ( G e. UPGraph -> I Fn dom I )
23	13	fneq2i 5451	. . . . . . . 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 16097	. . . . . 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 7163	. . . 4 |- ( ( ph /\ r e. A ) -> DECID ( I ` r ) = { U } )
30	29	ralrimiva 2615	. . 3 |- ( ph -> A. r e. A DECID ( I ` r ) = { U } )
31		fveqeq2 5679	. . . . 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 2778	. . 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 7197	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 2203   A.wral 2520   {crab 2524   C_ wss 3211   ~Pcpw 3669   {csn 3689   class class class wbr 4109   dom cdm 4749   Fn wfn 5347   ` cfv 5352   1oc1o 6640   2oc2o 6641   ~~ cen 6973   Fincfn 6975  Vtxcvtx 16007  iEdgciedg 16008  UPGraphcupgr 16086

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-fin 6978  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-upgren 16088

This theorem is referenced by:  vtxdgfifival  16286  vtxdgfif  16288  vtxdfifiun  16292  vtxd0nedgbfi  16294  vtxduspgrfvedgfi  16296