Theorem uspgrloopvtx 29548

Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29281). (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 6910	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5442	. . 3 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4		opvtxfv 29036	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = 𝑉)
5	3, 4	mpan2 691	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = 𝑉)
6	2, 5	eqtrid 2787	1 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ⟨cop 4637  ‘cfv 6563  Vtxcvtx 29028

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-1st 8013  df-vtx 29030

This theorem is referenced by:  uspgrloopvtxel  29549  uspgrloopnb0  29552  uspgrloopvd2  29553