Theorem mapdordlem1 41745

Description: Lemma for mapdord 41747. (Contributed by NM, 27-Jan-2015.)

Hypothesis

Ref	Expression
mapdordlem1.t	⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌}

Assertion

Ref	Expression
mapdordlem1	⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))

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

Allowed substitution hint:   𝑇(𝑔)

Proof of Theorem mapdordlem1

Step	Hyp	Ref	Expression
1		2fveq3 6827	. . . 4 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝐿‘𝑔)) = (𝑂‘(𝐿‘𝐽)))
2	1	fveq2d 6826	. . 3 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝑂‘(𝑂‘(𝐿‘𝐽))))
3	2	eleq1d 2816	. 2 ⊢ (𝑔 = 𝐽 → ((𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌 ↔ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
4		mapdordlem1.t	. 2 ⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌}
5	3, 4	elrab2 3645	1 ⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395  ‘cfv 6481

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489

This theorem is referenced by:  mapdordlem2  41746