Theorem opvtxfvi 28987

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 28982	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
4	1, 2, 3	mp2an 692	1 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Syntax hints:   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579  ‘cfv 6481  Vtxcvtx 28974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6437  df-fun 6483  df-fv 6489  df-1st 7921  df-vtx 28976

This theorem is referenced by:  graop  29007  vtxvalsnop  29019  uhgrspanop  29274  fusgrfis  29308  cusgrsize  29433  fusgrmaxsize  29443  vtxdgop  29449  vtxdginducedm1  29522  vtxdginducedm1fi  29523  finsumvtxdg2ssteplem4  29527  finsumvtxdg2size  29529  eupth2lem3  30216  konigsberglem1  30232  konigsberglem2  30233  konigsberglem3  30234