Theorem edgopval 28884

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	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Proof of Theorem edgopval

Step	Hyp	Ref	Expression
1		edgval 28882	. 2 ⊢ (Edg‘⟨𝑉, 𝐸⟩) = ran (iEdg‘⟨𝑉, 𝐸⟩)
2		opiedgfv 28840	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
3	2	rneqd 5944	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
4	1, 3	eqtrid 2780	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4638  ran crn 5683  ‘cfv 6553  iEdgciedg 28830  Edgcedg 28880

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 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746

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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  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 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-2nd 8000  df-iedg 28832  df-edg 28881

This theorem is referenced by:  edgov  28885  cusgrsize  29288  uspgrloopedg  29352  uspgrsprfo  47288