Theorem uspgrloopvtxel 29450

Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29182). (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 29449	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
3		eleq2 2818	. . . . 5 ⊢ (𝑉 = (Vtx‘𝐺) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝐺)))
4	3	biimpd 229	. . . 4 ⊢ (𝑉 = (Vtx‘𝐺) → (𝑁 ∈ 𝑉 → 𝑁 ∈ (Vtx‘𝐺)))
5	4	eqcoms 2738	. . 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 506	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4591  ⟨cop 4597  ‘cfv 6513  Vtxcvtx 28929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-iota 6466  df-fun 6515  df-fv 6521  df-1st 7970  df-vtx 28931

This theorem is referenced by: (None)