Theorem mapdordlem1bN 42005

Description: Lemma for mapdord 42008. (Contributed by NM, 27-Jan-2015.) (New usage is discouraged.)

Hypothesis

Ref	Expression
mapdordlem1b.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}

Assertion

Ref	Expression
mapdordlem1bN	⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))

Distinct variable groups:   𝑔,𝐹   𝑔,𝐽   𝑔,𝐿   𝑔,𝑂

Allowed substitution hint:   𝐶(𝑔)

Proof of Theorem mapdordlem1bN

Step	Hyp	Ref	Expression
1		mapdordlem1b.c	. 2 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
2	1	lcfl1lem 41861	1 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3401  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508

This theorem is referenced by: (None)