Theorem lpvtx 15959

Description: The endpoints of a loop (which is an edge at index J) are two (identical) vertices A. (Contributed by AV, 1-Feb-2021.)

Hypothesis

Ref	Expression
lpvtx.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
lpvtx	|- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> A e. (Vtx ` G ) )

Proof of Theorem lpvtx

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1023	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> G e. UHGraph )
2		lpvtx.i	. . . . . . 7 |- I = (iEdg ` G )
3	2	uhgrfun 15957	. . . . . 6 |- ( G e. UHGraph -> Fun I )
4	3	funfnd 5359	. . . . 5 |- ( G e. UHGraph -> I Fn dom I )
5	4	3ad2ant1 1044	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> I Fn dom I )
6		simp2 1024	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> J e. dom I )
7	2	uhgrm 15958	. . . 4 |- ( ( G e. UHGraph /\ I Fn dom I /\ J e. dom I ) -> E. j j e. ( I ` J ) )
8	1, 5, 6, 7	syl3anc 1273	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> E. j j e. ( I ` J ) )
9		eleq2 2294	. . . . 5 |- ( ( I ` J ) = { A } -> ( j e. ( I ` J ) <-> j e. { A } ) )
10	9	exbidv 1872	. . . 4 |- ( ( I ` J ) = { A } -> ( E. j j e. ( I ` J ) <-> E. j j e. { A } ) )
11	10	3ad2ant3 1046	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( E. j j e. ( I ` J ) <-> E. j j e. { A } ) )
12	8, 11	mpbid 147	. 2 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> E. j j e. { A } )
13		eqid 2230	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
14	13, 2	uhgrss 15955	. . . . 5 |- ( ( G e. UHGraph /\ J e. dom I ) -> ( I ` J ) C_ (Vtx ` G ) )
15	14	3adant3 1043	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( I ` J ) C_ (Vtx ` G ) )
16		sseq1 3249	. . . . 5 |- ( ( I ` J ) = { A } -> ( ( I ` J ) C_ (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
17	16	3ad2ant3 1046	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( ( I ` J ) C_ (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
18	15, 17	mpbid 147	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> { A } C_ (Vtx ` G ) )
19		snmb 3794	. . . 4 |- ( A e. _V <-> E. j j e. { A } )
20		snssg 3808	. . . 4 |- ( A e. _V -> ( A e. (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
21	19, 20	sylbir 135	. . 3 |- ( E. j j e. { A } -> ( A e. (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
22	18, 21	syl5ibrcom 157	. 2 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( E. j j e. { A } -> A e. (Vtx ` G ) ) )
23	12, 22	mpd 13	1 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> A e. (Vtx ` G ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2201   _Vcvv 2801   C_ wss 3199   {csn 3670   dom cdm 4727   Fn wfn 5323   ` cfv 5328  Vtxcvtx 15892  iEdgciedg 15893  UHGraphcuhgr 15947

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-cnre 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-fo 5334  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-sub 8357  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-dec 9617  df-ndx 13108  df-slot 13109  df-base 13111  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-uhgrm 15949

This theorem is referenced by: (None)