Theorem bnj911 31002

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
bnj911.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj911.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj911	|- ( ( f Fn n /\ ph /\ ps ) -> A. i ( f Fn n /\ ph /\ ps ) )

Distinct variable groups:   f, i   i, n   ph, i

Allowed substitution hints:   ph( y, f, n)   ps( y, f, i, n)   A( y, f, i, n)   R( y, f, i, n)   X( y, f, i, n)

Proof of Theorem bnj911

Step	Hyp	Ref	Expression
1		bnj911.2	. . 3 |- ( 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	. 2 |- ( ps -> A. i ps )
3	2	bnj1350 30896	1 |- ( ( f Fn n /\ ph /\ ps ) -> A. i ( f Fn n /\ ph /\ ps ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   (/)c0 3915   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` 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-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

This theorem is referenced by:  bnj916  31003  bnj1014  31030