Theorem bnj965 31012

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.)

Hypotheses

Ref	Expression
bnj965.1	|- ( ps <-> A. i e. om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj965.2	|- ( ps" <-> [. G / f ]. ps )
bnj965.12000	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj965.13000	|- G = ( f u. { <. n , C >. } )

Assertion

Ref	Expression
bnj965	|- ( ps" <-> A. i e. om ( suc i e. N -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )

Distinct variable groups:   A, f   i, G   f, N   R, f   f, i, y   y, n

Allowed substitution hints:   ps( y, f, i, m, n)   A( y, i, m, n)   C( y, f, i, m, n)   R( y, i, m, n)   G( y, f, m, n)   N( y, i, m, n)   ps"( y, f, i, m, n)

Proof of Theorem bnj965

Step	Hyp	Ref	Expression
1		bnj965.1	. 2 |- ( ps <-> A. i e. om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
2		bnj965.2	. 2 |- ( ps" <-> [. G / f ]. ps )
3		bnj965.13000	. . 3 |- G = ( f u. { <. n , C >. } )
4	3	bnj918 30836	. 2 |- G e. _V
5		bnj965.12000	. 2 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
6	1, 2, 4, 5, 3	bnj1000 31011	1 |- ( ps" <-> A. i e. om ( suc i e. N -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   [.wsbc 3435   u. cun 3572   {csn 4177   <.cop 4183   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

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-sbc 3436  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-iun 4522  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj964  31013  bnj999  31027