Theorem opvtxfvi 29041

Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as function value. (Contributed by AV, 4-Mar-2021.)

Hypotheses

Ref	Expression
opvtxfvi.v	⊢ 𝑉 ∈ V
opvtxfvi.e	⊢ 𝐸 ∈ V

Assertion

Ref	Expression
opvtxfvi	⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Proof of Theorem opvtxfvi

Step	Hyp	Ref	Expression
1		opvtxfvi.v	. 2 ⊢ 𝑉 ∈ V
2		opvtxfvi.e	. 2 ⊢ 𝐸 ∈ V
3		opvtxfv 29036	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
4	1, 2, 3	mp2an 692	1 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Syntax hints:   = wceq 1537   ∈ wcel 2106  Vcvv 3478  ⟨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:  graop  29061  vtxvalsnop  29073  uhgrspanop  29328  fusgrfis  29362  cusgrsize  29487  fusgrmaxsize  29497  vtxdgop  29503  vtxdginducedm1  29576  vtxdginducedm1fi  29577  finsumvtxdg2ssteplem4  29581  finsumvtxdg2size  29583  eupth2lem3  30265  konigsberglem1  30281  konigsberglem2  30282  konigsberglem3  30283