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Theorem lpvtx 26295
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 1167 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐺 ∈ UHGraph)
2 lpvtx.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
32uhgrfun 26293 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
4 funfn 6129 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
53, 4sylib 210 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
653ad2ant1 1164 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼)
7 simp2 1168 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼)
82uhgrn0 26294 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼𝐽 ∈ dom 𝐼) → (𝐼𝐽) ≠ ∅)
91, 6, 7, 8syl3anc 1491 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ≠ ∅)
10 neeq1 3031 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅))
1110biimpd 221 . . . 4 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
12113ad2ant3 1166 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
139, 12mpd 15 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ≠ ∅)
14 eqid 2797 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
1514, 2uhgrss 26291 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
16153adant3 1163 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
17 sseq1 3820 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
18173ad2ant3 1166 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
1916, 18mpbid 224 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺))
20 snnzb 4441 . . . 4 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
21 snssg 4501 . . . 4 (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2220, 21sylbir 227 . . 3 ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2319, 22syl5ibrcom 239 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ({𝐴} ≠ ∅ → 𝐴 ∈ (Vtx‘𝐺)))
2413, 23mpd 15 1 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐴 ∈ (Vtx‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  wne 2969  Vcvv 3383  wss 3767  c0 4113  {csn 4366  dom cdm 5310  Fun wfun 6093   Fn wfn 6094  cfv 6099  Vtxcvtx 26223  iEdgciedg 26224  UHGraphcuhgr 26283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-uhgr 26285
This theorem is referenced by:  lppthon  27487  lp1cycl  27488
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