Theorem bnj1033 31037

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
bnj1033.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1033.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1033.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1033.4	|- ( th <-> ( R FrSe A /\ X e. A ) )
bnj1033.5	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
bnj1033.6	|- ( et <-> z e. trCl ( X , A , R ) )
bnj1033.7	|- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
bnj1033.8	|- D = ( om \ { (/) } )
bnj1033.9	|- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1033.10	|- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )

Assertion

Ref	Expression
bnj1033	|- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

Distinct variable groups:   A, f, i, n, y   z, A, f, i, n   z, B   D, i   R, f, i, n, y   z, R   f, X, i, n, y   z, X   ta, f, i, n, z   th, f, i, n, z   ph, i

Allowed substitution hints:   ph( y, z, f, n)   ps( y, z, f, i, n)   ch( y, z, f, i, n)   th( y)   ta( y)   et( y, z, f, i, n)   ze( y, z, f, i, n)   B( y, f, i, n)   D( y, z, f, n)   K( y, z, f, i, n)

Proof of Theorem bnj1033

Step	Hyp	Ref	Expression
1		bnj1033.1	. . . . 5 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj1033.2	. . . . 5 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj1033.8	. . . . 5 |- D = ( om \ { (/) } )
4		bnj1033.9	. . . . 5 |- K = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5		bnj1033.3	. . . . 5 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
6	1, 2, 3, 4, 5	bnj983 31021	. . . 4 |- ( z e. trCl ( X , A , R ) <-> E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) )
7		19.42v 1918	. . . . . . . . . 10 |- ( E. i ( ( th /\ ta ) /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
8		df-3an 1039	. . . . . . . . . . 11 |- ( ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
9	8	exbii 1774	. . . . . . . . . 10 |- ( E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> E. i ( ( th /\ ta ) /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
10		df-3an 1039	. . . . . . . . . 10 |- ( ( th /\ ta /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
11	7, 9, 10	3bitr4i 292	. . . . . . . . 9 |- ( E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( th /\ ta /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
12	11	exbii 1774	. . . . . . . 8 |- ( E. n E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> E. n ( th /\ ta /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
13		19.42v 1918	. . . . . . . . 9 |- ( E. n ( ( th /\ ta ) /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
14	10	exbii 1774	. . . . . . . . 9 |- ( E. n ( th /\ ta /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> E. n ( ( th /\ ta ) /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
15		df-3an 1039	. . . . . . . . 9 |- ( ( th /\ ta /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
16	13, 14, 15	3bitr4i 292	. . . . . . . 8 |- ( E. n ( th /\ ta /\ E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( th /\ ta /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
17	12, 16	bitri 264	. . . . . . 7 |- ( E. n E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( th /\ ta /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
18	17	exbii 1774	. . . . . 6 |- ( E. f E. n E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> E. f ( th /\ ta /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
19		19.42v 1918	. . . . . . 7 |- ( E. f ( ( th /\ ta ) /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
20	15	exbii 1774	. . . . . . 7 |- ( E. f ( th /\ ta /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> E. f ( ( th /\ ta ) /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
21		df-3an 1039	. . . . . . 7 |- ( ( th /\ ta /\ E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( ( th /\ ta ) /\ E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
22	19, 20, 21	3bitr4i 292	. . . . . 6 |- ( E. f ( th /\ ta /\ E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( th /\ ta /\ E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
23	18, 22	bitri 264	. . . . 5 |- ( E. f E. n E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) <-> ( th /\ ta /\ E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
24		bnj255 30771	. . . . . . . 8 |- ( ( th /\ ta /\ ch /\ ze ) <-> ( th /\ ta /\ ( ch /\ ze ) ) )
25		bnj1033.7	. . . . . . . . . . 11 |- ( ze <-> ( i e. n /\ z e. ( f ` i ) ) )
26	25	anbi2i 730	. . . . . . . . . 10 |- ( ( ch /\ ze ) <-> ( ch /\ ( i e. n /\ z e. ( f ` i ) ) ) )
27		3anass 1042	. . . . . . . . . 10 |- ( ( ch /\ i e. n /\ z e. ( f ` i ) ) <-> ( ch /\ ( i e. n /\ z e. ( f ` i ) ) ) )
28	26, 27	bitr4i 267	. . . . . . . . 9 |- ( ( ch /\ ze ) <-> ( ch /\ i e. n /\ z e. ( f ` i ) ) )
29	28	3anbi3i 1255	. . . . . . . 8 |- ( ( th /\ ta /\ ( ch /\ ze ) ) <-> ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
30	24, 29	bitri 264	. . . . . . 7 |- ( ( th /\ ta /\ ch /\ ze ) <-> ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
31	30	3exbii 1776	. . . . . 6 |- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) <-> E. f E. n E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) )
32		bnj1033.10	. . . . . 6 |- ( E. f E. n E. i ( th /\ ta /\ ch /\ ze ) -> z e. B )
33	31, 32	sylbir 225	. . . . 5 |- ( E. f E. n E. i ( th /\ ta /\ ( ch /\ i e. n /\ z e. ( f ` i ) ) ) -> z e. B )
34	23, 33	sylbir 225	. . . 4 |- ( ( th /\ ta /\ E. f E. n E. i ( ch /\ i e. n /\ z e. ( f ` i ) ) ) -> z e. B )
35	6, 34	syl3an3b 1364	. . 3 |- ( ( th /\ ta /\ z e. trCl ( X , A , R ) ) -> z e. B )
36	35	3expia 1267	. 2 |- ( ( th /\ ta ) -> ( z e. trCl ( X , A , R ) -> z e. B ) )
37	36	ssrdv 3609	1 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760   TrFow-bnj19 30762

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-in 3581  df-ss 3588  df-iun 4522  df-fn 5891  df-bnj17 30753  df-bnj18 30761

This theorem is referenced by:  bnj1034  31038