Theorem bnj540 30962

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
bnj540.1	|- ( ps <-> A. i e. om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj540.2	|- ( ps" <-> [. G / f ]. ps )
bnj540.3	|- G e. _V

Assertion

Ref	Expression
bnj540	|- ( 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   f, G, i, y   f, N   R, f

Allowed substitution hints:   ps( y, f, i)   A( y, i)   R( y, i)   N( y, i)   ps"( y, f, i)

Proof of Theorem bnj540

Step	Hyp	Ref	Expression
1		bnj540.2	. 2 |- ( ps" <-> [. G / f ]. ps )
2		bnj540.1	. . . 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 |- ( [. G / f ]. ps <-> [. G / f ]. A. i e. om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
4		bnj540.3	. . . 4 |- G e. _V
5	4	bnj538 30809	. . 3 |- ( [. G / f ]. A. i e. om ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om [. G / f ]. ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
6		sbcimg 3477	. . . . 5 |- ( G e. _V -> ( [. G / f ]. ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> ( [. G / f ]. suc i e. N -> [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
7	4, 6	ax-mp 5	. . . 4 |- ( [. G / f ]. ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> ( [. G / f ]. suc i e. N -> [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
8	7	ralbii 2980	. . 3 |- ( A. i e. om [. G / f ]. ( suc i e. N -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( [. G / f ]. suc i e. N -> [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
9	3, 5, 8	3bitri 286	. 2 |- ( [. G / f ]. ps <-> A. i e. om ( [. G / f ]. suc i e. N -> [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
10	4	bnj525 30807	. . . 4 |- ( [. G / f ]. suc i e. N <-> suc i e. N )
11		fveq1 6190	. . . . . 6 |- ( f = G -> ( f ` suc i ) = ( G ` suc i ) )
12		fveq1 6190	. . . . . . 7 |- ( f = G -> ( f ` i ) = ( G ` i ) )
13	12	bnj1113 30856	. . . . . 6 |- ( f = G -> U_ y e. ( f ` i ) pred ( y , A , R ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
14	11, 13	eqeq12d 2637	. . . . 5 |- ( f = G -> ( ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) <-> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
15	4, 14	sbcie 3470	. . . 4 |- ( [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) <-> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
16	10, 15	imbi12i 340	. . 3 |- ( ( [. G / f ]. suc i e. N -> [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> ( suc i e. N -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
17	16	ralbii 2980	. 2 |- ( A. i e. om ( [. G / f ]. suc i e. N -> [. G / f ]. ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. N -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
18	1, 9, 17	3bitri 286	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   _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-rex 2918  df-v 3202  df-sbc 3436  df-in 3581  df-ss 3588  df-uni 4437  df-iun 4522  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj580  30983  bnj607  30986