Theorem uspgrloopvtxel 29600

Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29332). (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 29599	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
3		eleq2 2826	. . . . 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 2745	. . 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 1542   ∈ wcel 2114  {csn 4568  ⟨cop 4574  ‘cfv 6492  Vtxcvtx 29079

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-1st 7935  df-vtx 29081

This theorem is referenced by: (None)