Theorem uspgrloopvtx 27882

Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 27616). (Contributed by AV, 17-Dec-2020.)

Hypothesis

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

Assertion

Ref	Expression
uspgrloopvtx	⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Proof of Theorem uspgrloopvtx

Step	Hyp	Ref	Expression
1		uspgrloopvtx.g	. . 3 ⊢ 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
2	1	fveq2i 6777	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5354	. . 3 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4		opvtxfv 27374	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = 𝑉)
5	3, 4	mpan2 688	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = 𝑉)
6	2, 5	eqtrid 2790	1 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567  ‘cfv 6433  Vtxcvtx 27366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831  df-vtx 27368

This theorem is referenced by:  uspgrloopvtxel  27883  uspgrloopnb0  27886  uspgrloopvd2  27887