Theorem vtxlpfi 16140

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 7055	. . . . . . . 8 |- ( ( V e. Fin /\ p e. V /\ q e. V ) -> DECID p = q )
7	3, 4, 5, 6	syl3anc 1273	. . . . . . 7 |- ( ( ph /\ ( p e. V /\ q e. V ) ) -> DECID p = q )
8	7	ralrimivva 2614	. . . . . 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 15949	. . . . . 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 15947	. . . . . . . . 9 |- ( G e. UPGraph -> I : dom I --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
22	21	ffnd 5483	. . . . . . . 8 |- ( G e. UPGraph -> I Fn dom I )
23	13	fneq2i 5425	. . . . . . . 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 15952	. . . . . 6 |- ( ( G e. UPGraph /\ I Fn A /\ r e. A ) -> ( I ` r ) e. Fin )
28	20, 25, 26, 27	syl3anc 1273	. . . . 5 |- ( ( ph /\ r e. A ) -> ( I ` r ) e. Fin )
29	9, 11, 19, 28	eqsndc 7094	. . . 4 |- ( ( ph /\ r e. A ) -> DECID ( I ` r ) = { U } )
30	29	ralrimiva 2605	. . 3 |- ( ph -> A. r e. A DECID ( I ` r ) = { U } )
31		fveqeq2 5648	. . . . 5 |- ( r = x -> ( ( I ` r ) = { U } <-> ( I ` x ) = { U } ) )
32	31	dcbid 845	. . . 4 |- ( r = x -> (DECID ( I ` r ) = { U } <-> DECID ( I ` x ) = { U } ) )
33	32	cbvralv 2767	. . 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 7128	1 |- ( ph -> { x e. A | ( I ` x ) = { U } } e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   A.wral 2510   {crab 2514   C_ wss 3200   ~Pcpw 3652   {csn 3669   class class class wbr 4088   dom cdm 4725   Fn wfn 5321   ` cfv 5326   1oc1o 6574   2oc2o 6575   ~~ cen 6906   Fincfn 6908  Vtxcvtx 15862  iEdgciedg 15863  UPGraphcupgr 15941

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-fin 6911  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-upgren 15943

This theorem is referenced by:  vtxdgfifival  16141  vtxdgfif  16143  vtxdfifiun  16147  vtxd0nedgbfi  16149  vtxduspgrfvedgfi  16151