Theorem lpvtx 25963

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

Step	Hyp	Ref	Expression
1		simp1 1061	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> G e. UHGraph )
2		lpvtx.i	. . . . . . 7 |- I = (iEdg ` G )
3	2	uhgrfun 25961	. . . . . 6 |- ( G e. UHGraph -> Fun I )
4		funfn 5918	. . . . . 6 |- ( Fun I <-> I Fn dom I )
5	3, 4	sylib 208	. . . . 5 |- ( G e. UHGraph -> I Fn dom I )
6	5	3ad2ant1 1082	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> I Fn dom I )
7		simp2 1062	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> J e. dom I )
8	2	uhgrn0 25962	. . . 4 |- ( ( G e. UHGraph /\ I Fn dom I /\ J e. dom I ) -> ( I ` J ) =/= (/) )
9	1, 6, 7, 8	syl3anc 1326	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( I ` J ) =/= (/) )
10		neeq1 2856	. . . . 5 |- ( ( I ` J ) = { A } -> ( ( I ` J ) =/= (/) <-> { A } =/= (/) ) )
11	10	biimpd 219	. . . 4 |- ( ( I ` J ) = { A } -> ( ( I ` J ) =/= (/) -> { A } =/= (/) ) )
12	11	3ad2ant3 1084	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( ( I ` J ) =/= (/) -> { A } =/= (/) ) )
13	9, 12	mpd 15	. 2 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> { A } =/= (/) )
14		eqid 2622	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
15	14, 2	uhgrss 25959	. . . . 5 |- ( ( G e. UHGraph /\ J e. dom I ) -> ( I ` J ) C_ (Vtx ` G ) )
16	15	3adant3 1081	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( I ` J ) C_ (Vtx ` G ) )
17		sseq1 3626	. . . . 5 |- ( ( I ` J ) = { A } -> ( ( I ` J ) C_ (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
18	17	3ad2ant3 1084	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( ( I ` J ) C_ (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
19	16, 18	mpbid 222	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> { A } C_ (Vtx ` G ) )
20		snnzb 4254	. . . 4 |- ( A e. _V <-> { A } =/= (/) )
21		snssg 4327	. . . 4 |- ( A e. _V -> ( A e. (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
22	20, 21	sylbir 225	. . 3 |- ( { A } =/= (/) -> ( A e. (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
23	19, 22	syl5ibrcom 237	. 2 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( { A } =/= (/) -> A e. (Vtx ` G ) ) )
24	13, 23	mpd 15	1 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> A e. (Vtx ` G ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   dom cdm 5114   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-uhgr 25953

This theorem is referenced by:  lppthon  27011  lp1cycl  27012