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Theorem edgopval 28044
Description: The edges of a graph represented as ordered pair. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.)
Assertion
Ref Expression
edgopval ((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Proof of Theorem edgopval
StepHypRef Expression
1 edgval 28042 . 2 (Edg‘⟨𝑉, 𝐸⟩) = ran (iEdg‘⟨𝑉, 𝐸⟩)
2 opiedgfv 28000 . . 3 ((𝑉𝑊𝐸𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
32rneqd 5898 . 2 ((𝑉𝑊𝐸𝑋) → ran (iEdg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
41, 3eqtrid 2789 1 ((𝑉𝑊𝐸𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cop 4597  ran crn 5639  cfv 6501  iEdgciedg 27990  Edgcedg 28040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-2nd 7927  df-iedg 27992  df-edg 28041
This theorem is referenced by:  edgov  28045  cusgrsize  28444  uspgrloopedg  28508  uspgrsprfo  46124
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