Theorem bnj864 30992

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
bnj864.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj864.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj864.3	|- D = ( om \ { (/) } )
bnj864.4	|- ( ch <-> ( R FrSe A /\ X e. A /\ n e. D ) )
bnj864.5	|- ( th <-> ( f Fn n /\ ph /\ ps ) )

Assertion

Ref	Expression
bnj864	|- ( ch -> E! f th )

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

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

Proof of Theorem bnj864

Step	Hyp	Ref	Expression
1		bnj864.1	. . . . 5 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj864.2	. . . . 5 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj864.3	. . . . 5 |- D = ( om \ { (/) } )
4	1, 2, 3	bnj852 30991	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) )
5		df-ral 2917	. . . . . 6 |- ( A. n e. D E! f ( f Fn n /\ ph /\ ps ) <-> A. n ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) )
6	5	imbi2i 326	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ X e. A ) -> A. n ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) ) )
7		19.21v 1868	. . . . 5 |- ( A. n ( ( R FrSe A /\ X e. A ) -> ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) ) <-> ( ( R FrSe A /\ X e. A ) -> A. n ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) ) )
8		impexp 462	. . . . . . 7 |- ( ( ( ( R FrSe A /\ X e. A ) /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ X e. A ) -> ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) ) )
9		df-3an 1039	. . . . . . . . 9 |- ( ( R FrSe A /\ X e. A /\ n e. D ) <-> ( ( R FrSe A /\ X e. A ) /\ n e. D ) )
10	9	bicomi 214	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ n e. D ) <-> ( R FrSe A /\ X e. A /\ n e. D ) )
11	10	imbi1i 339	. . . . . . 7 |- ( ( ( ( R FrSe A /\ X e. A ) /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) ) <-> ( ( R FrSe A /\ X e. A /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
12	8, 11	bitr3i 266	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) -> ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) ) <-> ( ( R FrSe A /\ X e. A /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
13	12	albii 1747	. . . . 5 |- ( A. n ( ( R FrSe A /\ X e. A ) -> ( n e. D -> E! f ( f Fn n /\ ph /\ ps ) ) ) <-> A. n ( ( R FrSe A /\ X e. A /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
14	6, 7, 13	3bitr2i 288	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) -> A. n e. D E! f ( f Fn n /\ ph /\ ps ) ) <-> A. n ( ( R FrSe A /\ X e. A /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
15	4, 14	mpbi 220	. . 3 |- A. n ( ( R FrSe A /\ X e. A /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) )
16	15	spi 2054	. 2 |- ( ( R FrSe A /\ X e. A /\ n e. D ) -> E! f ( f Fn n /\ ph /\ ps ) )
17		bnj864.4	. 2 |- ( ch <-> ( R FrSe A /\ X e. A /\ n e. D ) )
18		bnj864.5	. . 3 |- ( th <-> ( f Fn n /\ ph /\ ps ) )
19	18	eubii 2492	. 2 |- ( E! f th <-> E! f ( f Fn n /\ ph /\ ps ) )
20	16, 17, 19	3imtr4i 281	1 |- ( ch -> E! f th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   E!weu 2470   A.wral 2912   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   predc-bnj14 30754   FrSe w-bnj15 30758

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj849  30995