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Theorem lpvtx 26839
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 1133 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐺 ∈ UHGraph)
2 lpvtx.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
32uhgrfun 26837 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
43funfnd 6359 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
543ad2ant1 1130 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐼 Fn dom 𝐼)
6 simp2 1134 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → 𝐽 ∈ dom 𝐼)
72uhgrn0 26838 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐼 Fn dom 𝐼𝐽 ∈ dom 𝐼) → (𝐼𝐽) ≠ ∅)
81, 5, 6, 7syl3anc 1368 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ≠ ∅)
9 neeq1 3069 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ ↔ {𝐴} ≠ ∅))
109biimpd 232 . . . 4 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
11103ad2ant3 1132 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ≠ ∅ → {𝐴} ≠ ∅))
128, 11mpd 15 . 2 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ≠ ∅)
13 eqid 2821 . . . . . 6 (Vtx‘𝐺) = (Vtx‘𝐺)
1413, 2uhgrss 26835 . . . . 5 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
15143adant3 1129 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → (𝐼𝐽) ⊆ (Vtx‘𝐺))
16 sseq1 3968 . . . . 5 ((𝐼𝐽) = {𝐴} → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
17163ad2ant3 1132 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → ((𝐼𝐽) ⊆ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
1815, 17mpbid 235 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐽 ∈ dom 𝐼 ∧ (𝐼𝐽) = {𝐴}) → {𝐴} ⊆ (Vtx‘𝐺))
19 snnzb 4627 . . . 4 (𝐴 ∈ V ↔ {𝐴} ≠ ∅)
20 snssg 4690 . . . 4 (𝐴 ∈ V → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2119, 20sylbir 238 . . 3 ({𝐴} ≠ ∅ → (𝐴 ∈ (Vtx‘𝐺) ↔ {𝐴} ⊆ (Vtx‘𝐺)))
2218, 21syl5ibrcom 250 . 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 209  w3a 1084   = wceq 1538  wcel 2115  wne 3007  Vcvv 3471  wss 3910  c0 4266  {csn 4540  dom cdm 5528   Fn wfn 6323  cfv 6328  Vtxcvtx 26767  iEdgciedg 26768  UHGraphcuhgr 26827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-uhgr 26829
This theorem is referenced by:  lppthon  27914  lp1cycl  27915
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