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Theorem mapdordlem1 40811
Description: Lemma for mapdord 40813. (Contributed by NM, 27-Jan-2015.)
Hypothesis
Ref Expression
mapdordlem1.t 𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}
Assertion
Ref Expression
mapdordlem1 (𝐽𝑇 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
Distinct variable groups:   𝑔,𝐹   𝑔,𝐽   𝑔,𝐿   𝑔,𝑂   𝑔,𝑌
Allowed substitution hint:   𝑇(𝑔)

Proof of Theorem mapdordlem1
StepHypRef Expression
1 2fveq3 6897 . . . 4 (𝑔 = 𝐽 → (𝑂‘(𝐿𝑔)) = (𝑂‘(𝐿𝐽)))
21fveq2d 6896 . . 3 (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿𝑔))) = (𝑂‘(𝑂‘(𝐿𝐽))))
32eleq1d 2817 . 2 (𝑔 = 𝐽 → ((𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌 ↔ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
4 mapdordlem1.t . 2 𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}
53, 4elrab2 3687 1 (𝐽𝑇 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1540  wcel 2105  {crab 3431  cfv 6544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552
This theorem is referenced by:  mapdordlem2  40812
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