Theorem bnj953 31009

Description: Technical lemma for bnj69 31078. 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.)

Hypotheses

Ref	Expression
bnj953.1	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj953.2	|- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )

Assertion

Ref	Expression
bnj953	|- ( ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )

Proof of Theorem bnj953

Step	Hyp	Ref	Expression
1		bnj312 30778	. . 3 |- ( ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) <-> ( ( G ` suc i ) = ( f ` suc i ) /\ ( G ` i ) = ( f ` i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) )
2		bnj251 30768	. . 3 |- ( ( ( G ` suc i ) = ( f ` suc i ) /\ ( G ` i ) = ( f ` i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) <-> ( ( G ` suc i ) = ( f ` suc i ) /\ ( ( G ` i ) = ( f ` i ) /\ ( ( i e. om /\ suc i e. n ) /\ ps ) ) ) )
3	1, 2	bitri 264	. 2 |- ( ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) <-> ( ( G ` suc i ) = ( f ` suc i ) /\ ( ( G ` i ) = ( f ` i ) /\ ( ( i e. om /\ suc i e. n ) /\ ps ) ) ) )
4		bnj953.1	. . . . . 6 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
5	4	bnj115 30791	. . . . 5 |- ( ps <-> A. i ( ( i e. om /\ suc i e. n ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
6		sp 2053	. . . . . 6 |- ( A. i ( ( 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 ) ) )
7	6	impcom 446	. . . . 5 |- ( ( ( i e. om /\ suc i e. n ) /\ A. i ( ( i e. om /\ suc i e. n ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
8	5, 7	sylan2b 492	. . . 4 |- ( ( ( i e. om /\ suc i e. n ) /\ ps ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
9		bnj953.2	. . . . 5 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )
10	9	bnj956 30847	. . . 4 |- ( ( G ` i ) = ( f ` i ) -> U_ y e. ( G ` i ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
11		eqtr3 2643	. . . 4 |- ( ( ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) /\ U_ y e. ( G ` i ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) -> ( f ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
12	8, 10, 11	syl2anr 495	. . 3 |- ( ( ( G ` i ) = ( f ` i ) /\ ( ( i e. om /\ suc i e. n ) /\ ps ) ) -> ( f ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
13		eqtr 2641	. . 3 |- ( ( ( G ` suc i ) = ( f ` suc i ) /\ ( f ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
14	12, 13	sylan2 491	. 2 |- ( ( ( G ` suc i ) = ( f ` suc i ) /\ ( ( G ` i ) = ( f ` i ) /\ ( ( i e. om /\ suc i e. n ) /\ ps ) ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
15	3, 14	sylbi 207	1 |- ( ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   U_ciun 4520   suc csuc 5725   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   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-ral 2917  df-rex 2918  df-iun 4522  df-bnj17 30753

This theorem is referenced by:  bnj967  31015