Theorem umgr2v2evtx 29780

Description: The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.)

Hypothesis

Ref	Expression
umgr2v2evtx.g	⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩

Assertion

Ref	Expression
umgr2v2evtx	⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Proof of Theorem umgr2v2evtx

Step	Hyp	Ref	Expression
1		umgr2v2evtx.g	. . 3 ⊢ 𝐺 = ⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩
2	1	fveq2i 6874	. 2 ⊢ (Vtx‘𝐺) = (Vtx‘⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩)
3		prex 5400	. . 3 ⊢ {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} ∈ V
4		opvtxfv 29263	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩} ∈ V) → (Vtx‘⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩) = 𝑉)
5	3, 4	mpan2 703	. 2 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘⟨𝑉, {⟨0, {𝐴, 𝐵}⟩, ⟨1, {𝐴, 𝐵}⟩}⟩) = 𝑉)
6	2, 5	eqtrid 2812	1 ⊢ (𝑉 ∈ 𝑊 → (Vtx‘𝐺) = 𝑉)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457  {cpr 4587  ⟨cop 4591  ‘cfv 6525  0cc0 11088  1c1 11089  Vtxcvtx 29255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-1st 7974  df-vtx 29257

This theorem is referenced by:  umgr2v2evtxel  29781  umgr2v2e  29784  umgr2v2enb1  29785