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Theorem iedgvalOLD 25862
Description: Obsolete version of iedgval 25860 as of 11-Nov-2021. (Contributed by AV, 21-Sep-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
iedgvalOLD (𝐺𝑉 → (iEdg‘𝐺) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))

Proof of Theorem iedgvalOLD
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 elex 3207 . 2 (𝐺𝑉𝐺 ∈ V)
2 eleq1 2687 . . . 4 (𝑔 = 𝐺 → (𝑔 ∈ (V × V) ↔ 𝐺 ∈ (V × V)))
3 fveq2 6178 . . . 4 (𝑔 = 𝐺 → (2nd𝑔) = (2nd𝐺))
4 fveq2 6178 . . . 4 (𝑔 = 𝐺 → (.ef‘𝑔) = (.ef‘𝐺))
52, 3, 4ifbieq12d 4104 . . 3 (𝑔 = 𝐺 → if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)) = if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)))
6 df-iedg 25858 . . 3 iEdg = (𝑔 ∈ V ↦ if(𝑔 ∈ (V × V), (2nd𝑔), (.ef‘𝑔)))
7 fvex 6188 . . . 4 (2nd𝐺) ∈ V
8 fvex 6188 . . . 4 (.ef‘𝐺) ∈ V
97, 8ifex 4147 . . 3 if(𝐺 ∈ (V × V), (2nd𝐺), (.ef‘𝐺)) ∈ V
105, 6, 9fvmpt 6269 . 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 1481  wcel 1988  Vcvv 3195  ifcif 4077   × cxp 5102  cfv 5876  2nd c2nd 7152  .efcedgf 25848  iEdgciedg 25856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-iedg 25858
This theorem is referenced by:  funiedgdm2valOLD  25877  funiedgdmge2valOLD  25881
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