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Theorem iedgval 25678
 Description: The set of indexed edges of a graph. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
iedgval (𝐺𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))

Proof of Theorem iedgval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝐺𝑉𝐺 ∈ V)
2 eleq1 2676 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (2nd𝑔) = (2nd𝐺))
4 fveq2 6103 . . . 4 (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
52, 3, 4ifbieq12d 4063 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
6 df-iedg 25676 . . 3 iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))
7 fvex 6113 . . . 4 (2nd𝐺) ∈ V
8 fvex 6113 . . . 4 (.ef‘𝐺) ∈ V
97, 8ifex 4106 . . 3 if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) ∈ V
105, 6, 9fvmpt 6191 . 2 (𝐺 ∈ V → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
111, 10syl 17 1 (𝐺𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ifcif 4036   × cxp 5036  ‘cfv 5804  2nd c2nd 7058  .efcedgf 25667  iEdgciedg 25674 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-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 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-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-iota 5768  df-fun 5806  df-fv 5812  df-iedg 25676 This theorem is referenced by:  opiedgval  25683  funiedgdm2val  25689  funiedgdmge2val  25692  snstriedgval  25713  iedgval0  25715  iedgvalsnop  25717
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