Theorem opvtxfvi 28777

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 28772	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉)
4	1, 2, 3	mp2an 689	1 ⊢ (Vtx‘⟨𝑉, 𝐸⟩) = 𝑉

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629  ‘cfv 6537  Vtxcvtx 28764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fv 6545  df-1st 7974  df-vtx 28766

This theorem is referenced by:  graop  28797  vtxvalsnop  28809  uhgrspanop  29061  fusgrfis  29095  cusgrsize  29220  fusgrmaxsize  29230  vtxdgop  29236  vtxdginducedm1  29309  vtxdginducedm1fi  29310  finsumvtxdg2ssteplem4  29314  finsumvtxdg2size  29316  eupth2lem3  29998  konigsberglem1  30014  konigsberglem2  30015  konigsberglem3  30016