Theorem uspgrloopvtxel 28702

Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 28435). (Contributed by AV, 17-Dec-2020.)

Hypothesis

Ref	Expression
uspgrloopvtx.g	⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩

Assertion

Ref	Expression
uspgrloopvtxel	⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Proof of Theorem uspgrloopvtxel

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	. . 3 ⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
2	1	uspgrloopvtx 28701	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
3		eleq2 2822	. . . . 5 ⊢ (𝑉 = (Vtx‘𝐺) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝐺)))
4	3	biimpd 228	. . . 4 ⊢ (𝑉 = (Vtx‘𝐺) → (𝑁 ∈ 𝑉 → 𝑁 ∈ (Vtx‘𝐺)))
5	4	eqcoms 2740	. . 3 ⊢ ((Vtx‘𝐺) = 𝑉 → (𝑁 ∈ 𝑉 → 𝑁 ∈ (Vtx‘𝐺)))
6	5	com12 32	. 2 ⊢ (𝑁 ∈ 𝑉 → ((Vtx‘𝐺) = 𝑉 → 𝑁 ∈ (Vtx‘𝐺)))
7	2, 6	mpan9 507	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {csn 4623  ⟨cop 4629  ‘cfv 6533  Vtxcvtx 28185

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-iota 6485  df-fun 6535  df-fv 6541  df-1st 7959  df-vtx 28187

This theorem is referenced by: (None)