Theorem bnj981 31020

Description: Technical lemma for bnj69 31078. 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
bnj981.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj981.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj981.3	|- D = ( om \ { (/) } )
bnj981.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj981.5	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )

Assertion

Ref	Expression
bnj981	|- ( Z e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) )

Distinct variable groups:   A, f, i, n, y   D, i, y   R, f, i, n, y   f, X, i, n, y   f, Z, i, n, y   ph, i, y

Allowed substitution hints:   ph( f, n)   ps( y, f, i, n)   ch( y, f, i, n)   B( y, f, i, n)   D( f, n)

Proof of Theorem bnj981

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ y Z e. trCl ( X , A , R )
2		bnj981.2	. . . . . . . . . . . 12 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ y om
4		nfv 1843	. . . . . . . . . . . . . 14 |- F/ y suc i e. n
5		nfiu1 4550	. . . . . . . . . . . . . . 15 |- F/_ y U_ y e. ( f ` i ) pred ( y , A , R )
6	5	nfeq2 2780	. . . . . . . . . . . . . 14 |- F/ y ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R )
7	4, 6	nfim 1825	. . . . . . . . . . . . 13 |- F/ y ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
8	3, 7	nfral 2945	. . . . . . . . . . . 12 |- F/ y A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
9	2, 8	nfxfr 1779	. . . . . . . . . . 11 |- F/ y ps
10	9	nf5ri 2065	. . . . . . . . . 10 |- ( ps -> A. y ps )
11		bnj981.5	. . . . . . . . . 10 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
12	10, 11	bnj1096 30853	. . . . . . . . 9 |- ( ch -> A. y ch )
13	12	nf5i 2024	. . . . . . . 8 |- F/ y ch
14		nfv 1843	. . . . . . . 8 |- F/ y i e. n
15		nfv 1843	. . . . . . . 8 |- F/ y Z e. ( f ` i )
16	13, 14, 15	nf3an 1831	. . . . . . 7 |- F/ y ( ch /\ i e. n /\ Z e. ( f ` i ) )
17	16	nfex 2154	. . . . . 6 |- F/ y E. i ( ch /\ i e. n /\ Z e. ( f ` i ) )
18	17	nfex 2154	. . . . 5 |- F/ y E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) )
19	18	nfex 2154	. . . 4 |- F/ y E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) )
20	1, 19	nfim 1825	. . 3 |- F/ y ( Z e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) )
21		eleq1 2689	. . . 4 |- ( y = Z -> ( y e. trCl ( X , A , R ) <-> Z e. trCl ( X , A , R ) ) )
22		eleq1 2689	. . . . . 6 |- ( y = Z -> ( y e. ( f ` i ) <-> Z e. ( f ` i ) ) )
23	22	3anbi3d 1405	. . . . 5 |- ( y = Z -> ( ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) )
24	23	3exbidv 1853	. . . 4 |- ( y = Z -> ( E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) <-> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) )
25	21, 24	imbi12d 334	. . 3 |- ( y = Z -> ( ( y e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) ) <-> ( Z e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) ) )
26		bnj981.1	. . . 4 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
27		bnj981.3	. . . 4 |- D = ( om \ { (/) } )
28		bnj981.4	. . . 4 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
29	26, 2, 27, 28, 11	bnj917 31004	. . 3 |- ( y e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ y e. ( f ` i ) ) )
30	20, 25, 29	vtoclg1f 3265	. 2 |- ( Z e. trCl ( X , A , R ) -> ( Z e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) ) )
31	30	pm2.43i 52	1 |- ( Z e. trCl ( X , A , R ) -> E. f E. n E. i ( ch /\ i e. n /\ Z e. ( f ` i ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   trClc-bnj18 30760

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-iun 4522  df-fn 5891  df-bnj17 30753  df-bnj18 30761

This theorem is referenced by:  bnj1128  31058