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Theorem opvtxov 27665
Description: The set of vertices of a graph represented as an ordered pair of vertices and indexed edges as operation value. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
opvtxov ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → (𝑉Vtx𝐞) = 𝑉)

Proof of Theorem opvtxov
StepHypRef Expression
1 df-ov 7341 . 2 (𝑉Vtx𝐞) = (Vtx‘⟚𝑉, 𝐞⟩)
2 opvtxfv 27664 . 2 ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → (Vtx‘⟚𝑉, 𝐞⟩) = 𝑉)
31, 2eqtrid 2788 1 ((𝑉 ∈ 𝑋 ∧ 𝐞 ∈ 𝑌) → (𝑉Vtx𝐞) = 𝑉)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  âŸšcop 4580  â€˜cfv 6480  (class class class)co 7338  Vtxcvtx 27656
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5244  ax-nul 5251  ax-pr 5373  ax-un 7651
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-iota 6432  df-fun 6482  df-fv 6488  df-ov 7341  df-1st 7900  df-vtx 27658
This theorem is referenced by: (None)
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