Theorem bnj98 30937

Description: Technical lemma for bnj150 30946. 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.)

Assertion

Ref	Expression
bnj98	|- A. i e. om ( suc i e. 1o -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) )

Proof of Theorem bnj98

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- i e. _V
2	1	sucid 5804	. . . . 5 |- i e. suc i
3	2	n0ii 3922	. . . 4 |- -. suc i = (/)
4		df-suc 5729	. . . . . 6 |- suc i = ( i u. { i } )
5		df-un 3579	. . . . . 6 |- ( i u. { i } ) = { x | ( x e. i \/ x e. { i } ) }
6	4, 5	eqtri 2644	. . . . 5 |- suc i = { x | ( x e. i \/ x e. { i } ) }
7		df1o2 7572	. . . . . . 7 |- 1o = { (/) }
8	6, 7	eleq12i 2694	. . . . . 6 |- ( suc i e. 1o <-> { x | ( x e. i \/ x e. { i } ) } e. { (/) } )
9		elsni 4194	. . . . . 6 |- ( { x | ( x e. i \/ x e. { i } ) } e. { (/) } -> { x | ( x e. i \/ x e. { i } ) } = (/) )
10	8, 9	sylbi 207	. . . . 5 |- ( suc i e. 1o -> { x | ( x e. i \/ x e. { i } ) } = (/) )
11	6, 10	syl5eq 2668	. . . 4 |- ( suc i e. 1o -> suc i = (/) )
12	3, 11	mto 188	. . 3 |- -. suc i e. 1o
13	12	pm2.21i 116	. 2 |- ( suc i e. 1o -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) )
14	13	rgenw 2924	1 |- A. i e. om ( suc i e. 1o -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   u. cun 3572   (/)c0 3915   {csn 4177   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-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-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-suc 5729  df-1o 7560

This theorem is referenced by:  bnj150  30946