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Theorem mapdordlem1 41638
Description: Lemma for mapdord 41640. (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 6911 . . . 4 (𝑔 = 𝐽 → (𝑂‘(𝐿𝑔)) = (𝑂‘(𝐿𝐽)))
21fveq2d 6910 . . 3 (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿𝑔))) = (𝑂‘(𝑂‘(𝐿𝐽))))
32eleq1d 2826 . 2 (𝑔 = 𝐽 → ((𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌 ↔ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
4 mapdordlem1.t . 2 𝑇 = {𝑔𝐹 ∣ (𝑂‘(𝑂‘(𝐿𝑔))) ∈ 𝑌}
53, 4elrab2 3695 1 (𝐽𝑇 ↔ (𝐽𝐹 ∧ (𝑂‘(𝑂‘(𝐿𝐽))) ∈ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569
This theorem is referenced by:  mapdordlem2  41639
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