Theorem edgov 29069

Description: The edges of a graph represented as ordered pair, shown as operation value. Although a little less intuitive, this representation is often used because it is shorter than the representation as function value of a graph given as ordered pair, see edgopval 29068. The representation ran 𝐸 for the set of edges is even shorter, though. (Contributed by AV, 2-Jan-2020.) (Revised by AV, 13-Oct-2020.)

Assertion

Ref	Expression
edgov	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Proof of Theorem edgov

Step	Hyp	Ref	Expression
1		df-ov 7434	. 2 ⊢ (𝑉Edg𝐸) = (Edg‘⟨𝑉, 𝐸⟩)
2		edgopval 29068	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
3	1, 2	eqtrid 2789	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ⟨cop 4632  ran crn 5686  ‘cfv 6561  (class class class)co 7431  Edgcedg 29064

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-2nd 8015  df-iedg 29016  df-edg 29065

This theorem is referenced by: (None)