Theorem edgov 28864

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 28863. 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 7423	. 2 ⊢ (𝑉Edg𝐸) = (Edg‘⟨𝑉, 𝐸⟩)
2		edgopval 28863	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
3	1, 2	eqtrid 2780	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ⟨cop 4635  ran crn 5679  ‘cfv 6548  (class class class)co 7420  Edgcedg 28859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-2nd 7994  df-iedg 28811  df-edg 28860

This theorem is referenced by: (None)