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Theorem edgov 28980
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 28979. 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 ((𝑉𝑊𝐸𝑋) → (𝑉Edg𝐸) = ran 𝐸)

Proof of Theorem edgov
StepHypRef Expression
1 df-ov 7426 . 2 (𝑉Edg𝐸) = (Edg‘⟨𝑉, 𝐸⟩)
2 edgopval 28979 . 2 ((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
31, 2eqtrid 2777 1 ((𝑉𝑊𝐸𝑋) → (𝑉Edg𝐸) = ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4638  ran crn 5682  cfv 6553  (class class class)co 7423  Edgcedg 28975
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 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432  ax-un 7745
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7426  df-2nd 8003  df-iedg 28927  df-edg 28976
This theorem is referenced by: (None)
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