Theorem bnj155 30949

Description: Technical lemma for bnj153 30950. 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
bnj155.1	|- ( ps1 <-> [. g / f ]. ps' )
bnj155.2	|- ( ps' <-> A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj155	|- ( ps1 <-> A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )

Distinct variable groups:   A, f   R, f   f, g, i, y

Allowed substitution hints:   A( y, g, i)   R( y, g, i)   ps'( y, f, g, i)   ps1( y, f, g, i)

Proof of Theorem bnj155

Step	Hyp	Ref	Expression
1		bnj155.1	. 2 |- ( ps1 <-> [. g / f ]. ps' )
2		bnj155.2	. . 3 |- ( ps' <-> A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3	2	sbcbii 3491	. 2 |- ( [. g / f ]. ps' <-> [. g / f ]. A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
4		vex 3203	. . 3 |- g e. _V
5		fveq1 6190	. . . . . 6 |- ( f = g -> ( f ` suc i ) = ( g ` suc i ) )
6		fveq1 6190	. . . . . . 7 |- ( f = g -> ( f ` i ) = ( g ` i ) )
7	6	iuneq1d 4545	. . . . . 6 |- ( f = g -> U_ y e. ( f ` i ) pred ( y , A , R ) = U_ y e. ( g ` i ) pred ( y , A , R ) )
8	5, 7	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 ) ) )
9	8	imbi2d 330	. . . 4 |- ( f = g -> ( ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) )
10	9	ralbidv 2986	. . 3 |- ( f = g -> ( A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) ) )
11	4, 10	sbcie 3470	. 2 |- ( [. g / f ]. A. i e. om ( suc i e. 1o -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. 1o -> ( g ` suc i ) = U_ y e. ( g ` i ) pred ( y , A , R ) ) )
12	1, 3, 11	3bitri 286	1 |- ( ps1 <-> A. i e. om ( suc i e. 1o -> ( 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_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

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:  bnj153  30950