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Theorem edgval 25868
 Description: The edges of a graph. (Contributed by AV, 1-Jan-2020.)
Assertion
Ref Expression
edgval (𝐺𝑉 → (Edges‘𝐺) = ran (2nd𝐺))

Proof of Theorem edgval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-edg 25865 . . 3 Edges = (𝑔 ∈ V ↦ ran (2nd𝑔))
21a1i 11 . 2 (𝐺𝑉 → Edges = (𝑔 ∈ V ↦ ran (2nd𝑔)))
3 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (2nd𝑔) = (2nd𝐺))
43adantl 481 . . 3 ((𝐺𝑉𝑔 = 𝐺) → (2nd𝑔) = (2nd𝐺))
54rneqd 5274 . 2 ((𝐺𝑉𝑔 = 𝐺) → ran (2nd𝑔) = ran (2nd𝐺))
6 elex 3185 . 2 (𝐺𝑉𝐺 ∈ V)
7 fvex 6113 . . . 4 (2nd𝐺) ∈ V
87rnex 6992 . . 3 ran (2nd𝐺) ∈ V
98a1i 11 . 2 (𝐺𝑉 → ran (2nd𝐺) ∈ V)
102, 5, 6, 9fvmptd 6197 1 (𝐺𝑉 → (Edges‘𝐺) = ran (2nd𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  2nd c2nd 7058  Edgescedg 25860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-edg 25865 This theorem is referenced by:  edgopval  25869  edgss  25881  edgprvtx  25914  usgrafiedg  25945  0eusgraiff0rgracl  26468
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