Theorem bnj917 31004

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
bnj917.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj917.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj917.3	|- D = ( om \ { (/) } )
bnj917.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj917.5	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )

Assertion

Ref	Expression
bnj917	|- ( y e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )

Distinct variable groups:   A, f, i, n, y   D, i   R, f, i, n, y   f, X, i, n, y   ph, i

Allowed substitution hints:   ph( y, f, n)   ps( y, f, i, n)   ch( y, f, i, n)   B( y, f, i, n)   D( y, f, n)

Proof of Theorem bnj917

Step	Hyp	Ref	Expression
1		bnj917.1	. . 3 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj917.2	. . 3 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj917.3	. . 3 |- D = ( om \ { (/) } )
4		bnj917.4	. . 3 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5		biid 251	. . 3 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ph /\ ps ) )
6	1, 2, 3, 4, 5	bnj916 31003	. 2 |- ( y e. trCl ( X , A , R ) -> E. f E. n E. i ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
7		bnj917.5	. . . . . 6 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
8		bnj252 30769	. . . . . 6 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
9	7, 8	bitri 264	. . . . 5 |- ( ch <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) )
10	9	3anbi1i 1253	. . . 4 |- ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) /\ i e. n /\ y e. ( f ` i ) ) )
11		bnj253 30770	. . . 4 |- ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) <-> ( ( n e. D /\ ( f Fn n /\ ph /\ ps ) ) /\ i e. n /\ y e. ( f ` i ) ) )
12	10, 11	bitr4i 267	. . 3 |- ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
13	12	3exbii 1776	. 2 |- ( E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> E. f E. n E. i ( n e. D /\ ( f Fn n /\ ph /\ ps ) /\ i e. n /\ y e. ( f ` i ) ) )
14	6, 13	sylibr 224	1 |- ( y e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   trClc-bnj18 30760

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-v 3202  df-iun 4522  df-fn 5891  df-bnj17 30753  df-bnj18 30761

This theorem is referenced by:  bnj981  31020  bnj996  31025