Theorem bnj126 30943

Description: Technical lemma for bnj150 30946. 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
bnj126.1	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj126.2	|- ( ps' <-> [. 1o / n ]. ps )
bnj126.3	|- ( ps" <-> [. F / f ]. ps' )
bnj126.4	|- F = { <. (/) , pred ( x , A , R ) >. }

Assertion

Ref	Expression
bnj126	|- ( ps" <-> A. i e. om ( suc i e. 1o -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )

Distinct variable groups:   A, f, n   f, F, i, y   R, f, n   i, n, y

Allowed substitution hints:   ps( x, y, f, i, n)   A( x, y, i)   R( x, y, i)   F( x, n)   ps'( x, y, f, i, n)   ps"( x, y, f, i, n)

Proof of Theorem bnj126

Step	Hyp	Ref	Expression
1		bnj126.3	. 2 |- ( ps" <-> [. F / f ]. ps' )
2		bnj126.2	. . 3 |- ( ps' <-> [. 1o / n ]. ps )
3	2	sbcbii 3491	. 2 |- ( [. F / f ]. ps' <-> [. F / f ]. [. 1o / n ]. ps )
4		bnj126.1	. . 3 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
5		bnj126.4	. . . 4 |- F = { <. (/) , pred ( x , A , R ) >. }
6	5	bnj95 30934	. . 3 |- F e. _V
7	4, 6	bnj106 30938	. 2 |- ( [. F / f ]. [. 1o / n ]. ps <-> A. i e. om ( suc i e. 1o -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
8	1, 3, 7	3bitri 286	1 |- ( ps" <-> A. i e. om ( suc i e. 1o -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   [.wsbc 3435   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   suc csuc 5725   ` cfv 5888   omcom 7065   1oc1o 7553   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  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-pw 4160  df-sn 4178  df-pr 4180  df-uni 4437  df-iun 4522  df-br 4654  df-suc 5729  df-iota 5851  df-fv 5896  df-1o 7560

This theorem is referenced by:  bnj150  30946  bnj153  30950