Theorem bnj1021 31034

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
bnj1021.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1021.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1021.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1021.4	|- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
bnj1021.5	|- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
bnj1021.6	|- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
bnj1021.13	|- D = ( om \ { (/) } )
bnj1021.14	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }

Assertion

Ref	Expression
bnj1021	|- E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )

Distinct variable groups:   A, f, i, n, y   D, i   R, f, i, n, y   f, X, i, n, y   ch, m, p   et, m, p   th, f, i, n   ph, i   m, n, th, p

Allowed substitution hints:   ph( y, z, f, m, n, p)   ps( y, z, f, i, m, n, p)   ch( y, z, f, i, n)   th( y, z)   ta( y, z, f, i, m, n, p)   et( y, z, f, i, n)   A( z, m, p)   B( y, z, f, i, m, n, p)   D( y, z, f, m, n, p)   R( z, m, p)   X( z, m, p)

Proof of Theorem bnj1021

Step	Hyp	Ref	Expression
1		bnj1021.1	. . . 4 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj1021.2	. . . 4 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj1021.3	. . . 4 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4		bnj1021.4	. . . 4 |- ( th <-> ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) /\ z e. pred ( y , A , R ) ) )
5		bnj1021.5	. . . 4 |- ( ta <-> ( m e. om /\ n = suc m /\ p = suc n ) )
6		bnj1021.6	. . . 4 |- ( et <-> ( i e. n /\ y e. ( f ` i ) ) )
7		bnj1021.13	. . . 4 |- D = ( om \ { (/) } )
8		bnj1021.14	. . . 4 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
9	1, 2, 3, 4, 5, 6, 7, 8	bnj996 31025	. . 3 |- E. f E. n E. i E. m E. p ( th -> ( ch /\ ta /\ et ) )
10		anclb 570	. . . . . 6 |- ( ( th -> ( ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ( ch /\ ta /\ et ) ) ) )
11		bnj252 30769	. . . . . . 7 |- ( ( th /\ ch /\ ta /\ et ) <-> ( th /\ ( ch /\ ta /\ et ) ) )
12	11	imbi2i 326	. . . . . 6 |- ( ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ( ch /\ ta /\ et ) ) ) )
13	10, 12	bitr4i 267	. . . . 5 |- ( ( th -> ( ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ ta /\ et ) ) )
14	13	2exbii 1775	. . . 4 |- ( E. m E. p ( th -> ( ch /\ ta /\ et ) ) <-> E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) )
15	14	3exbii 1776	. . 3 |- ( E. f E. n E. i E. m E. p ( th -> ( ch /\ ta /\ et ) ) <-> E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) )
16	9, 15	mpbi 220	. 2 |- E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) )
17		19.37v 1910	. . . . 5 |- ( E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> E. p ( th /\ ch /\ ta /\ et ) ) )
18		bnj1019 30850	. . . . . 6 |- ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )
19	18	imbi2i 326	. . . . 5 |- ( ( th -> E. p ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
20	17, 19	bitri 264	. . . 4 |- ( E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
21	20	2exbii 1775	. . 3 |- ( E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
22	21	2exbii 1775	. 2 |- ( E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
23	16, 22	mpbi 220	1 |- E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )

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   \ 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   FrSe w-bnj15 30758   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-fn 5891  df-om 7066  df-bnj17 30753  df-bnj18 30761

This theorem is referenced by:  bnj907  31035