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Theorem edgov 26944
 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 26943. 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 7153 . 2 (𝑉Edg𝐸) = (Edg‘⟨𝑉, 𝐸⟩)
2 edgopval 26943 . 2 ((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
31, 2syl5eq 2805 1 ((𝑉𝑊𝐸𝑋) → (𝑉Edg𝐸) = ran 𝐸)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4528  ran crn 5525  ‘cfv 6335  (class class class)co 7150  Edgcedg 26939 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-2nd 7694  df-iedg 26891  df-edg 26940 This theorem is referenced by: (None)
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