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Theorem lpvtx 28328
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 1137 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐺 ∈ UHGraph)
2 lpvtx.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
32uhgrfun 28326 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
43funfnd 6580 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
543ad2ant1 1134 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼)
6 simp2 1138 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼)
72uhgrn0 28327 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼𝐽 ∈ dom 𝐼) → (𝐼𝐽) ≠ ∅)
81, 5, 6, 7syl3anc 1372 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ≠ ∅)
9 neeq1 3004 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅))
109biimpd 228 . . . 4 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
11103ad2ant3 1136 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
128, 11mpd 15 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ≠ ∅)
13 eqid 2733 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
1413, 2uhgrss 28324 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
15143adant3 1133 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
16 sseq1 4008 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
17163ad2ant3 1136 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
1815, 17mpbid 231 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺))
19 snnzb 4723 . . . 4 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
20 snssg 4788 . . . 4 (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2119, 20sylbir 234 . . 3 ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2218, 21syl5ibrcom 246 . 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 205  w3a 1088   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  wss 3949  c0 4323  {csn 4629  dom cdm 5677   Fn wfn 6539  cfv 6544  Vtxcvtx 28256  iEdgciedg 28257  UHGraphcuhgr 28316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-uhgr 28318
This theorem is referenced by:  lppthon  29404  lp1cycl  29405
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