Theorem bnj1021 29823

Description: Technical lemma for bnj69 29867. 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 29814	. . 3 |- E. f E. n E. i E. m E. p ( th -> ( ch /\ ta /\ et ) )
10		anclb 554	. . . . . 6 |- ( ( th -> ( ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ( ch /\ ta /\ et ) ) ) )
11		bnj252 29556	. . . . . . 7 |- ( ( th /\ ch /\ ta /\ et ) <-> ( th /\ ( ch /\ ta /\ et ) ) )
12	11	imbi2i 318	. . . . . 6 |- ( ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ( ch /\ ta /\ et ) ) ) )
13	10, 12	bitr4i 260	. . . . 5 |- ( ( th -> ( ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ ta /\ et ) ) )
14	13	2exbii 1729	. . . 4 |- ( E. m E. p ( th -> ( ch /\ ta /\ et ) ) <-> E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) ) )
15	14	3exbii 1730	. . 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 213	. 2 |- E. f E. n E. i E. m E. p ( th -> ( th /\ ch /\ ta /\ et ) )
17		19.37v 1836	. . . . 5 |- ( E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> E. p ( th /\ ch /\ ta /\ et ) ) )
18		bnj1019 29639	. . . . . 6 |- ( E. p ( th /\ ch /\ ta /\ et ) <-> ( th /\ ch /\ et /\ E. p ta ) )
19	18	imbi2i 318	. . . . 5 |- ( ( th -> E. p ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
20	17, 19	bitri 257	. . . 4 |- ( E. p ( th -> ( th /\ ch /\ ta /\ et ) ) <-> ( th -> ( th /\ ch /\ et /\ E. p ta ) ) )
21	20	2exbii 1729	. . 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 1729	. 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 213	1 |- E. f E. n E. i E. m ( th -> ( th /\ ch /\ et /\ E. p ta ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1454   E.wex 1673   e. wcel 1897   {cab 2447   A.wral 2748   E.wrex 2749   \ cdif 3412   (/)c0 3742   {csn 3979   U_ciun 4291   suc csuc 5443   Fn wfn 5595   ` cfv 5600   omcom 6718   /\ w-bnj17 29539   predc-bnj14 29541   FrSe w-bnj15 29545   trClc-bnj18 29547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-tr 4511  df-eprel 4763  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-fn 5603  df-om 6719  df-bnj17 29540  df-bnj18 29548

This theorem is referenced by:  bnj907  29824