Theorem lpvtx 15844

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 1002	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> G e. UHGraph )
2		lpvtx.i	. . . . . . 7 |- I = (iEdg ` G )
3	2	uhgrfun 15842	. . . . . 6 |- ( G e. UHGraph -> Fun I )
4	3	funfnd 5325	. . . . 5 |- ( G e. UHGraph -> I Fn dom I )
5	4	3ad2ant1 1023	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> I Fn dom I )
6		simp2 1003	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> J e. dom I )
7	2	uhgrm 15843	. . . 4 |- ( ( G e. UHGraph /\ I Fn dom I /\ J e. dom I ) -> E. j j e. ( I ` J ) )
8	1, 5, 6, 7	syl3anc 1252	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> E. j j e. ( I ` J ) )
9		eleq2 2273	. . . . 5 |- ( ( I ` J ) = { A } -> ( j e. ( I ` J ) <-> j e. { A } ) )
10	9	exbidv 1851	. . . 4 |- ( ( I ` J ) = { A } -> ( E. j j e. ( I ` J ) <-> E. j j e. { A } ) )
11	10	3ad2ant3 1025	. . 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 2209	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
14	13, 2	uhgrss 15840	. . . . 5 |- ( ( G e. UHGraph /\ J e. dom I ) -> ( I ` J ) C_ (Vtx ` G ) )
15	14	3adant3 1022	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( I ` J ) C_ (Vtx ` G ) )
16		sseq1 3227	. . . . 5 |- ( ( I ` J ) = { A } -> ( ( I ` J ) C_ (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
17	16	3ad2ant3 1025	. . . 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 3767	. . . 4 |- ( A e. _V <-> E. j j e. { A } )
20		snssg 3781	. . . 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 983   = wceq 1375   E.wex 1518   e. wcel 2180   _Vcvv 2779   C_ wss 3177   {csn 3646   dom cdm 4696   Fn wfn 5289   ` cfv 5294  Vtxcvtx 15778  iEdgciedg 15779  UHGraphcuhgr 15832

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-1cn 8060  ax-1re 8061  ax-icn 8062  ax-addcl 8063  ax-addrcl 8064  ax-mulcl 8065  ax-addcom 8067  ax-mulcom 8068  ax-addass 8069  ax-mulass 8070  ax-distr 8071  ax-i2m1 8072  ax-1rid 8074  ax-0id 8075  ax-rnegex 8076  ax-cnre 8078

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-fo 5300  df-fv 5302  df-riota 5927  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-sub 8287  df-inn 9079  df-2 9137  df-3 9138  df-4 9139  df-5 9140  df-6 9141  df-7 9142  df-8 9143  df-9 9144  df-n0 9338  df-dec 9547  df-ndx 13001  df-slot 13002  df-base 13004  df-edgf 15771  df-vtx 15780  df-iedg 15781  df-uhgrm 15834

This theorem is referenced by: (None)