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Theorem mapdordlem1 39912
Description: Lemma for mapdord 39914. (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 6830 . . . 4 (𝑔 = 𝐽 → (𝑂‘(𝐿𝑔)) = (𝑂‘(𝐿𝐽)))
21fveq2d 6829 . . 3 (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿𝑔))) = (𝑂‘(𝑂‘(𝐿𝐽))))
32eleq1d 2821 . 2 (𝑔 = 𝐽 → ((𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌 ↔ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
4 mapdordlem1.t . 2 𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}
53, 4elrab2 3637 1 (𝐽𝑇 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1540  wcel 2105  {crab 3403  cfv 6479
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 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487
This theorem is referenced by:  mapdordlem2  39913
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