Theorem lpvtx 16123

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 1024	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> G e. UHGraph )
2		lpvtx.i	. . . . . . 7 |- I = (iEdg ` G )
3	2	uhgrfun 16121	. . . . . 6 |- ( G e. UHGraph -> Fun I )
4	3	funfnd 5385	. . . . 5 |- ( G e. UHGraph -> I Fn dom I )
5	4	3ad2ant1 1045	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> I Fn dom I )
6		simp2 1025	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> J e. dom I )
7	2	uhgrm 16122	. . . 4 |- ( ( G e. UHGraph /\ I Fn dom I /\ J e. dom I ) -> E. j j e. ( I ` J ) )
8	1, 5, 6, 7	syl3anc 1274	. . 3 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> E. j j e. ( I ` J ) )
9		eleq2 2298	. . . . 5 |- ( ( I ` J ) = { A } -> ( j e. ( I ` J ) <-> j e. { A } ) )
10	9	exbidv 1874	. . . 4 |- ( ( I ` J ) = { A } -> ( E. j j e. ( I ` J ) <-> E. j j e. { A } ) )
11	10	3ad2ant3 1047	. . 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 2234	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
14	13, 2	uhgrss 16119	. . . . 5 |- ( ( G e. UHGraph /\ J e. dom I ) -> ( I ` J ) C_ (Vtx ` G ) )
15	14	3adant3 1044	. . . 4 |- ( ( G e. UHGraph /\ J e. dom I /\ ( I ` J ) = { A } ) -> ( I ` J ) C_ (Vtx ` G ) )
16		sseq1 3263	. . . . 5 |- ( ( I ` J ) = { A } -> ( ( I ` J ) C_ (Vtx ` G ) <-> { A } C_ (Vtx ` G ) ) )
17	16	3ad2ant3 1047	. . . 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 3815	. . . 4 |- ( A e. _V <-> E. j j e. { A } )
20		snssg 3830	. . . 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 1005   = wceq 1398   E.wex 1541   e. wcel 2205   _Vcvv 2815   C_ wss 3213   {csn 3691   dom cdm 4751   Fn wfn 5349   ` cfv 5354  Vtxcvtx 16056  iEdgciedg 16057  UHGraphcuhgr 16111

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8451  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-5 9304  df-6 9305  df-7 9306  df-8 9307  df-9 9308  df-n0 9502  df-dec 9716  df-ndx 13236  df-slot 13237  df-base 13239  df-edgf 16049  df-vtx 16058  df-iedg 16059  df-uhgrm 16113

This theorem is referenced by: (None)