Theorem mapdordlem1 36925

Description: Lemma for mapdord 36927. (Contributed by NM, 27-Jan-2015.)

Hypothesis

Ref	Expression
mapdordlem1.t	|- T = { g e. F | ( O ` ( O ` ( L ` g ) ) ) e. Y }

Assertion

Ref	Expression
mapdordlem1	|- ( J e. T <-> ( J e. F /\ ( O ` ( O ` ( L ` J ) ) ) e. 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		fveq2 6191	. . . . 5 |- ( g = J -> ( L ` g ) = ( L ` J ) )
2	1	fveq2d 6195	. . . 4 |- ( g = J -> ( O ` ( L ` g ) ) = ( O ` ( L ` J ) ) )
3	2	fveq2d 6195	. . 3 |- ( g = J -> ( O ` ( O ` ( L ` g ) ) ) = ( O ` ( O ` ( L ` J ) ) ) )
4	3	eleq1d 2686	. 2 |- ( g = J -> ( ( O ` ( O ` ( L ` g ) ) ) e. Y <-> ( O ` ( O ` ( L ` J ) ) ) e. Y ) )
5		mapdordlem1.t	. 2 |- T = { g e. F | ( O ` ( O ` ( L ` g ) ) ) e. Y }
6	4, 5	elrab2 3366	1 |- ( J e. T <-> ( J e. F /\ ( O ` ( O ` ( L ` J ) ) ) e. Y ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  mapdordlem2  36926