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Theorem lpvtx 26780
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 ∧ (𝐼𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))

Proof of Theorem lpvtx
StepHypRef Expression
1 simp1 1128 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐺 ∈ UHGraph)
2 lpvtx.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
32uhgrfun 26778 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
43funfnd 6379 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
543ad2ant1 1125 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼)
6 simp2 1129 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼)
72uhgrn0 26779 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼𝐽 ∈ dom 𝐼) → (𝐼𝐽) ≠ ∅)
81, 5, 6, 7syl3anc 1363 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ≠ ∅)
9 neeq1 3075 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅))
109biimpd 230 . . . 4 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
11103ad2ant3 1127 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
128, 11mpd 15 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ≠ ∅)
13 eqid 2818 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
1413, 2uhgrss 26776 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
15143adant3 1124 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
16 sseq1 3989 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
17163ad2ant3 1127 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
1815, 17mpbid 233 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺))
19 snnzb 4646 . . . 4 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
20 snssg 4709 . . . 4 (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2119, 20sylbir 236 . . 3 ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2218, 21syl5ibrcom 248 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ({𝐴} ≠ ∅ → 𝐴 ∈ (Vtx‘𝐺)))
2312, 22mpd 15 1 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  wss 3933  c0 4288  {csn 4557  dom cdm 5548   Fn wfn 6343  cfv 6348  Vtxcvtx 26708  iEdgciedg 26709  UHGraphcuhgr 26768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-uhgr 26770
This theorem is referenced by:  lppthon  27857  lp1cycl  27858
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