Theorem bnj1093 31048

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
bnj1093.1	|- E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B )
bnj1093.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1093.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )

Assertion

Ref	Expression
bnj1093	|- ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )

Distinct variable groups:   ch, j   ta, i   th, i   ta, j   th, j   D, i   f, i   i, n   ph, i

Allowed substitution hints:   ph( y, f, j, n)   ps( y, f, i, j, n)   ch( y, f, i, n)   th( y, f, n)   ta( y, f, n)   ze( y, f, i, j, n)   A( y, f, i, j, n)   B( y, f, i, j, n)   D( y, f, j, n)   R( y, f, i, j, n)   ph0( y, f, i, j, n)

Proof of Theorem bnj1093

Step	Hyp	Ref	Expression
1		bnj1093.2	. . . . . 6 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
2	1	bnj1095 30852	. . . . 5 |- ( ps -> A. i ps )
3		bnj1093.3	. . . . 5 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4	2, 3	bnj1096 30853	. . . 4 |- ( ch -> A. i ch )
5	4	bnj1350 30896	. . 3 |- ( ( th /\ ta /\ ch ) -> A. i ( th /\ ta /\ ch ) )
6		bnj1093.1	. . . . 5 |- E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B )
7		impexp 462	. . . . . 6 |- ( ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B ) <-> ( ( th /\ ta /\ ch ) -> ( ph0 -> ( f ` i ) C_ B ) ) )
8	7	exbii 1774	. . . . 5 |- ( E. j ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` i ) C_ B ) <-> E. j ( ( th /\ ta /\ ch ) -> ( ph0 -> ( f ` i ) C_ B ) ) )
9	6, 8	mpbi 220	. . . 4 |- E. j ( ( th /\ ta /\ ch ) -> ( ph0 -> ( f ` i ) C_ B ) )
10	9	19.37iv 1911	. . 3 |- ( ( th /\ ta /\ ch ) -> E. j ( ph0 -> ( f ` i ) C_ B ) )
11	5, 10	alrimih 1751	. 2 |- ( ( th /\ ta /\ ch ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )
12	11	bnj721 30827	1 |- ( ( th /\ ta /\ ch /\ ze ) -> A. i E. j ( ph0 -> ( f ` i ) C_ B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   C_ wss 3574   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` 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-10 2019  ax-12 2047

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-ral 2917  df-bnj17 30753

This theorem is referenced by:  bnj1030  31055