Theorem uspgrloopvtxel 27292

Description: A vertex in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27025). (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 27291	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)
3		eleq2 2901	. . . . 5 ⊢ (𝑉 = (Vtx‘𝐺) → (𝑁 ∈ 𝑉 ↔ 𝑁 ∈ (Vtx‘𝐺)))
4	3	biimpd 231	. . . 4 ⊢ (𝑉 = (Vtx‘𝐺) → (𝑁 ∈ 𝑉 → 𝑁 ∈ (Vtx‘𝐺)))
5	4	eqcoms 2829	. . 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 509	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑁 ∈ 𝑉) → 𝑁 ∈ (Vtx‘𝐺))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {csn 4561  ⟨cop 4567  ‘cfv 6350  Vtxcvtx 26775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-iota 6309  df-fun 6352  df-fv 6358  df-1st 7683  df-vtx 26777

This theorem is referenced by: (None)