Theorem bnj590 30980

Description: Technical lemma for bnj852 30991. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj590.1	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj590	|- ( ( B = suc i /\ ps ) -> ( i e. om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )

Proof of Theorem bnj590

Step	Hyp	Ref	Expression
1		bnj590.1	. . . 4 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
2		rsp 2929	. . . 4 |- ( A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) -> ( i e. om -> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
3	1, 2	sylbi 207	. . 3 |- ( ps -> ( i e. om -> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
4		eleq1 2689	. . . . 5 |- ( B = suc i -> ( B e. n <-> suc i e. n ) )
5		fveq2 6191	. . . . . 6 |- ( B = suc i -> ( f ` B ) = ( f ` suc i ) )
6	5	eqeq1d 2624	. . . . 5 |- ( B = suc i -> ( ( f ` B ) = U_ y e. ( f ` i ) pred ( y , A , R ) <-> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
7	4, 6	imbi12d 334	. . . 4 |- ( B = suc i -> ( ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
8	7	imbi2d 330	. . 3 |- ( B = suc i -> ( ( i e. om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) <-> ( i e. om -> ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
9	3, 8	syl5ibr 236	. 2 |- ( B = suc i -> ( ps -> ( i e. om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
10	9	imp 445	1 |- ( ( B = suc i /\ ps ) -> ( i e. om -> ( B e. n -> ( f ` B ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   U_ciun 4520   suc csuc 5725   ` cfv 5888   omcom 7065   predc-bnj14 30754

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-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:  bnj594  30982