Theorem rusgrnumwwlkslem 26864

Description: Lemma for rusgrnumwwlks 26869. (Contributed by Alexander van der Vekens, 23-Aug-2018.)

Assertion

Ref	Expression
rusgrnumwwlkslem	|- ( Y e. { w e. Z | ( w ` 0 ) = P } -> { w e. X | ( ph /\ ps ) } = { w e. X | ( ph /\ ( Y ` 0 ) = P /\ ps ) } )

Distinct variable groups:   w, P   w, Y   w, Z

Allowed substitution hints:   ph( w)   ps( w)   X( w)

Proof of Theorem rusgrnumwwlkslem

Step	Hyp	Ref	Expression
1		fveq1 6190	. . . 4 |- ( w = Y -> ( w ` 0 ) = ( Y ` 0 ) )
2	1	eqeq1d 2624	. . 3 |- ( w = Y -> ( ( w ` 0 ) = P <-> ( Y ` 0 ) = P ) )
3	2	elrab 3363	. 2 |- ( Y e. { w e. Z | ( w ` 0 ) = P } <-> ( Y e. Z /\ ( Y ` 0 ) = P ) )
4		ibar 525	. . . . 5 |- ( ( Y ` 0 ) = P -> ( ( ph /\ ps ) <-> ( ( Y ` 0 ) = P /\ ( ph /\ ps ) ) ) )
5		3anass 1042	. . . . . 6 |- ( ( ( Y ` 0 ) = P /\ ph /\ ps ) <-> ( ( Y ` 0 ) = P /\ ( ph /\ ps ) ) )
6		3ancoma 1045	. . . . . 6 |- ( ( ( Y ` 0 ) = P /\ ph /\ ps ) <-> ( ph /\ ( Y ` 0 ) = P /\ ps ) )
7	5, 6	bitr3i 266	. . . . 5 |- ( ( ( Y ` 0 ) = P /\ ( ph /\ ps ) ) <-> ( ph /\ ( Y ` 0 ) = P /\ ps ) )
8	4, 7	syl6bb 276	. . . 4 |- ( ( Y ` 0 ) = P -> ( ( ph /\ ps ) <-> ( ph /\ ( Y ` 0 ) = P /\ ps ) ) )
9	8	ad2antlr 763	. . 3 |- ( ( ( Y e. Z /\ ( Y ` 0 ) = P ) /\ w e. X ) -> ( ( ph /\ ps ) <-> ( ph /\ ( Y ` 0 ) = P /\ ps ) ) )
10	9	rabbidva 3188	. 2 |- ( ( Y e. Z /\ ( Y ` 0 ) = P ) -> { w e. X | ( ph /\ ps ) } = { w e. X | ( ph /\ ( Y ` 0 ) = P /\ ps ) } )
11	3, 10	sylbi 207	1 |- ( Y e. { w e. Z | ( w ` 0 ) = P } -> { w e. X | ( ph /\ ps ) } = { w e. X | ( ph /\ ( Y ` 0 ) = P /\ ps ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   ` cfv 5888   0cc0 9936

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  rusgrnumwwlks  26869