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Theorem lpvtx 25876
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 1059 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐺 ∈ UHGraph )
2 lpvtx.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
32uhgrfun 25874 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
4 funfn 5882 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
53, 4sylib 208 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
653ad2ant1 1080 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼)
7 simp2 1060 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼)
82uhgrn0 25875 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼𝐽 ∈ dom 𝐼) → (𝐼𝐽) ≠ ∅)
91, 6, 7, 8syl3anc 1323 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ≠ ∅)
10 neeq1 2852 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅))
1110biimpd 219 . . . 4 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
12113ad2ant3 1082 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
139, 12mpd 15 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ≠ ∅)
14 eqid 2621 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
1514, 2uhgrss 25872 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
16153adant3 1079 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
17 sseq1 3610 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
18173ad2ant3 1082 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
1916, 18mpbid 222 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺))
20 snnzb 4229 . . . 4 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
21 snssg 4301 . . . 4 (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2220, 21sylbir 225 . . 3 ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2319, 22syl5ibrcom 237 . 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 196  w3a 1036   = wceq 1480  wcel 1987  wne 2790  Vcvv 3189  wss 3559  c0 3896  {csn 4153  dom cdm 5079  Fun wfun 5846   Fn wfn 5847  cfv 5852  Vtxcvtx 25791  iEdgciedg 25792   UHGraph cuhgr 25864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-uhgr 25866
This theorem is referenced by:  lppthon  26894  lp1cycl  26895
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