Theorem bnj1039 31039

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
bnj1039.1	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1039.2	|- ( ps' <-> [. j / i ]. ps )

Assertion

Ref	Expression
bnj1039	|- ( ps' <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )

Proof of Theorem bnj1039

Step	Hyp	Ref	Expression
1		bnj1039.2	. 2 |- ( ps' <-> [. j / i ]. ps )
2		vex 3203	. . 3 |- j e. _V
3		bnj1039.1	. . . . 5 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
4		nfra1 2941	. . . . 5 |- F/ i A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
5	3, 4	nfxfr 1779	. . . 4 |- F/ i ps
6	5	sbcgf 3501	. . 3 |- ( j e. _V -> ( [. j / i ]. ps <-> ps ) )
7	2, 6	ax-mp 5	. 2 |- ( [. j / i ]. ps <-> ps )
8	1, 7, 3	3bitri 286	1 |- ( ps' <-> A. i e. om ( suc i e. n -> ( 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   _Vcvv 3200   [.wsbc 3435   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-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-ral 2917  df-v 3202  df-sbc 3436

This theorem is referenced by:  bnj1128  31058