Theorem edgopval 29310

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 29308	. 2 ⊢ (Edg‘⟨𝑉, 𝐸⟩) = ran (iEdg‘⟨𝑉, 𝐸⟩)
2		opiedgfv 29266	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (iEdg‘⟨𝑉, 𝐸⟩) = 𝐸)
3	2	rneqd 5919	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → ran (iEdg‘⟨𝑉, 𝐸⟩) = ran 𝐸)
4	1, 3	eqtrid 2812	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (Edg‘⟨𝑉, 𝐸⟩) = ran 𝐸)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ⟨cop 4591  ran crn 5653  ‘cfv 6525  iEdgciedg 29256  Edgcedg 29306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fv 6533  df-2nd 7975  df-iedg 29258  df-edg 29307

This theorem is referenced by:  edgov  29311  cusgrsize  29713  uspgrloopedg  29777  uspgrsprfo  48768