Theorem bnj222 30953

Description: Technical lemma for bnj229 30954. 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.)

Hypothesis

Ref	Expression
bnj222.1	|- ( ps <-> A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj222	|- ( ps <-> A. m e. om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )

Distinct variable groups:   A, i, m   i, F, m, y   i, N, m   R, i, m

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

Proof of Theorem bnj222

Step	Hyp	Ref	Expression
1		bnj222.1	. 2 |- ( ps <-> A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
2		suceq 5790	. . . . 5 |- ( i = m -> suc i = suc m )
3	2	eleq1d 2686	. . . 4 |- ( i = m -> ( suc i e. N <-> suc m e. N ) )
4	2	fveq2d 6195	. . . . 5 |- ( i = m -> ( F ` suc i ) = ( F ` suc m ) )
5		fveq2 6191	. . . . . 6 |- ( i = m -> ( F ` i ) = ( F ` m ) )
6	5	bnj1113 30856	. . . . 5 |- ( i = m -> U_ y e. ( F ` i ) pred ( y , A , R ) = U_ y e. ( F ` m ) pred ( y , A , R ) )
7	4, 6	eqeq12d 2637	. . . 4 |- ( i = m -> ( ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) <-> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
8	3, 7	imbi12d 334	. . 3 |- ( i = m -> ( ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) ) )
9	8	cbvralv 3171	. 2 |- ( A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) <-> A. m e. om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
10	1, 9	bitri 264	1 |- ( ps <-> A. m e. om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   A.wral 2912   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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-suc 5729  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj229  30954  bnj589  30979