Theorem opiedgov 28979

Description: The set of indexed edges 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
opiedgov	⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉iEdg𝐸) = 𝐸)

Proof of Theorem opiedgov

Step	Hyp	Ref	Expression
1		df-ov 7344	. 2 ⊢ (𝑉iEdg𝐸) = (iEdg‘⟨𝑉, 𝐸⟩)
2		opiedgfv 28978	. 2 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
3	1, 2	eqtrid 2777	1 ⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝑉iEdg𝐸) = 𝐸)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ⟨cop 4580  ‘cfv 6477  (class class class)co 7341  iEdgciedg 28968

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 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-2nd 7917  df-iedg 28970

This theorem is referenced by: (None)