Theorem bnj539 30961

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
bnj539.1	|- ( ps <-> A. i e. om ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
bnj539.2	|- ( ps' <-> [. M / n ]. ps )
bnj539.3	|- M e. _V

Assertion

Ref	Expression
bnj539	|- ( ps' <-> A. i e. om ( suc i e. M -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )

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

Allowed substitution hints:   ps( y, i, n)   A( y, i)   R( y, i)   F( y, i)   M( y, n)   ps'( y, i, n)

Proof of Theorem bnj539

Step	Hyp	Ref	Expression
1		bnj539.2	. 2 |- ( ps' <-> [. M / n ]. ps )
2		bnj539.1	. . . 4 |- ( ps <-> A. i e. om ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
3	2	sbcbii 3491	. . 3 |- ( [. M / n ]. ps <-> [. M / n ]. A. i e. om ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
4		bnj539.3	. . . . 5 |- M e. _V
5	4	bnj538 30809	. . . 4 |- ( [. M / n ]. A. i e. om ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> A. i e. om [. M / n ]. ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
6		sbcimg 3477	. . . . . . 7 |- ( M e. _V -> ( [. M / n ]. ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> ( [. M / n ]. suc i e. n -> [. M / n ]. ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) ) )
7	4, 6	ax-mp 5	. . . . . 6 |- ( [. M / n ]. ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> ( [. M / n ]. suc i e. n -> [. M / n ]. ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
8		sbcel2gv 3496	. . . . . . . 8 |- ( M e. _V -> ( [. M / n ]. suc i e. n <-> suc i e. M ) )
9	4, 8	ax-mp 5	. . . . . . 7 |- ( [. M / n ]. suc i e. n <-> suc i e. M )
10	4	bnj525 30807	. . . . . . 7 |- ( [. M / n ]. ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) <-> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) )
11	9, 10	imbi12i 340	. . . . . 6 |- ( ( [. M / n ]. suc i e. n -> [. M / n ]. ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> ( suc i e. M -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
12	7, 11	bitri 264	. . . . 5 |- ( [. M / n ]. ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> ( suc i e. M -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
13	12	ralbii 2980	. . . 4 |- ( A. i e. om [. M / n ]. ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. M -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
14	5, 13	bitri 264	. . 3 |- ( [. M / n ]. A. i e. om ( suc i e. n -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. M -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
15	3, 14	bitri 264	. 2 |- ( [. M / n ]. ps <-> A. i e. om ( suc i e. M -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
16	1, 15	bitri 264	1 |- ( ps' <-> A. i e. om ( suc i e. M -> ( 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-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-v 3202  df-sbc 3436

This theorem is referenced by:  bnj600  30989  bnj908  31001  bnj964  31013  bnj999  31027