Theorem mapdordlem1 38774

Description: Lemma for mapdord 38776. (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 6677	. . . 4 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝐿‘𝑔)) = (𝑂‘(𝐿‘𝐽)))
2	1	fveq2d 6676	. . 3 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝑂‘(𝑂‘(𝐿‘𝐽))))
3	2	eleq1d 2899	. 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 3685	1 ⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3144  ‘cfv 6357

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365

This theorem is referenced by:  mapdordlem2  38775