Theorem uspgrloopvtx 29846

Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29580). (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 6886	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3		snex 5412	. . 3 ⊢ {⟨𝐴, {𝑁}⟩} ∈ V
4		opvtxfv 29335	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = 𝑉)
5	3, 4	mpan2 703	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = 𝑉)
6	2, 5	eqtrid 2810	1 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4590  ⟨cop 4596  ‘cfv 6538  Vtxcvtx 29327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 7987  df-vtx 29329

This theorem is referenced by:  uspgrloopvtxel  29847  uspgrloopnb0  29850  uspgrloopvd2  29851