Theorem lpvtx 16203

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 16201	. . . . . 6 |- ( G e. UHGraph -> Fun I )
4	3	funfnd 5388	. . . . 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 16202	. . . 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 16199	. . . . 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 3265	. . . . 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 3818	. . . 4 |- ( A e. _V <-> E. j j e. { A } )
20		snssg 3833	. . . 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 3214   {csn 3694   dom cdm 4754   Fn wfn 5352   ` cfv 5357  Vtxcvtx 16136  iEdgciedg 16137  UHGraphcuhgr 16191

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 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-uhgrm 16193

This theorem is referenced by: (None)