Theorem opvtxfvi 28954

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 28949	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
4	1, 2, 3	mp2an 692	1 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Syntax hints:   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ⟨cop 4583  ‘cfv 6482  Vtxcvtx 28941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6438  df-fun 6484  df-fv 6490  df-1st 7924  df-vtx 28943

This theorem is referenced by:  graop  28974  vtxvalsnop  28986  uhgrspanop  29241  fusgrfis  29275  cusgrsize  29400  fusgrmaxsize  29410  vtxdgop  29416  vtxdginducedm1  29489  vtxdginducedm1fi  29490  finsumvtxdg2ssteplem4  29494  finsumvtxdg2size  29496  eupth2lem3  30180  konigsberglem1  30196  konigsberglem2  30197  konigsberglem3  30198