Theorem lpvtx 26847

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 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))

Proof of Theorem lpvtx

Step	Hyp	Ref	Expression
1		simp1 1132	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐺 ∈ UHGraph)
2		lpvtx.i	. . . . . . 7 ⊢ 𝐼 = (iEdg‘𝐺)
3	2	uhgrfun 26845	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
4	3	funfnd 6381	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
5	4	3ad2ant1 1129	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼)
6		simp2 1133	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼)
7	2	uhgrn0 26846	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝐽 ∈ dom 𝐼) → (𝐼‘𝐽) ≠ ∅)
8	1, 5, 6, 7	syl3anc 1367	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → (𝐼‘𝐽) ≠ ∅)
9		neeq1 3078	. . . . 5 ⊢ ((𝐼‘𝐽) = {𝐴} → ((𝐼‘𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅))
10	9	biimpd 231	. . . 4 ⊢ ((𝐼‘𝐽) = {𝐴} → ((𝐼‘𝐽) ≠ ∅ → {𝐴} ≠ ∅))
11	10	3ad2ant3 1131	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ((𝐼‘𝐽) ≠ ∅ → {𝐴} ≠ ∅))
12	8, 11	mpd 15	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → {𝐴} ≠ ∅)
13		eqid 2821	. . . . . 6 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13, 2	uhgrss 26843	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼‘𝐽) ⊆ (Vtx‘𝐺))
15	14	3adant3 1128	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → (𝐼‘𝐽) ⊆ (Vtx‘𝐺))
16		sseq1 3992	. . . . 5 ⊢ ((𝐼‘𝐽) = {𝐴} → ((𝐼‘𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
17	16	3ad2ant3 1131	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ((𝐼‘𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
18	15, 17	mpbid 234	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺))
19		snnzb 4648	. . . 4 ⊢ (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
20		snssg 4711	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
21	19, 20	sylbir 237	. . 3 ⊢ ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
22	18, 21	syl5ibrcom 249	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → ({𝐴} ≠ ∅ → 𝐴 ∈ (Vtx‘𝐺)))
23	12, 22	mpd 15	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼‘𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  Vcvv 3495   ⊆ wss 3936  ∅c0 4291  {csn 4561  dom cdm 5550   Fn wfn 6345  ‘cfv 6350  Vtxcvtx 26775  iEdgciedg 26776  UHGraphcuhgr 26835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-uhgr 26837

This theorem is referenced by:  lppthon  27924  lp1cycl  27925