Theorem bnj151 30947

Description: Technical lemma for bnj153 30950. 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
bnj151.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj151.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj151.3	|- D = ( om \ { (/) } )
bnj151.4	|- ( th <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj151.5	|- ( ta <-> A. m e. D ( m _E n -> [. m / n ]. th ) )
bnj151.6	|- ( ze <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
bnj151.7	|- ( ph' <-> [. 1o / n ]. ph )
bnj151.8	|- ( ps' <-> [. 1o / n ]. ps )
bnj151.9	|- ( th' <-> [. 1o / n ]. th )
bnj151.10	|- ( th0 <-> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn 1o /\ ph' /\ ps' ) ) )
bnj151.11	|- ( th1 <-> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) ) )
bnj151.12	|- ( ze' <-> [. 1o / n ]. ze )
bnj151.13	|- F = { <. (/) , pred ( x , A , R ) >. }
bnj151.14	|- ( ph" <-> [. F / f ]. ph' )
bnj151.15	|- ( ps" <-> [. F / f ]. ps' )
bnj151.16	|- ( ze" <-> [. F / f ]. ze' )
bnj151.17	|- ( ze0 <-> ( f Fn 1o /\ ph' /\ ps' ) )
bnj151.18	|- ( ze1 <-> [. g / f ]. ze0 )
bnj151.19	|- ( ph1 <-> [. g / f ]. ph' )
bnj151.20	|- ( ps1 <-> [. g / f ]. ps' )

Assertion

Ref	Expression
bnj151	|- ( n = 1o -> ( ( n e. D /\ ta ) -> th ) )

Distinct variable groups:   A, f, g, x   A, n, f, x   f, F, i, y   R, f, g, x   R, n   f, ze1   f, ze"   g, ze0   i, n, y   m, n

Allowed substitution hints:   ph( x, y, f, g, i, m, n)   ps( x, y, f, g, i, m, n)   th( x, y, f, g, i, m, n)   ta( x, y, f, g, i, m, n)   ze( x, y, f, g, i, m, n)   A( y, i, m)   D( x, y, f, g, i, m, n)   R( y, i, m)   F( x, g, m, n)   ph'( x, y, f, g, i, m, n)   ps'( x, y, f, g, i, m, n)   th'( x, y, f, g, i, m, n)   ze'( x, y, f, g, i, m, n)   ph"( x, y, f, g, i, m, n)   ps"( x, y, f, g, i, m, n)   ze"( x, y, g, i, m, n)   th0( x, y, f, g, i, m, n)   ze0( x, y, f, i, m, n)   ph1( x, y, f, g, i, m, n)   ps1( x, y, f, g, i, m, n)   th1( x, y, f, g, i, m, n)   ze1( x, y, g, i, m, n)

Proof of Theorem bnj151

Step	Hyp	Ref	Expression
1		bnj151.1	. . . . . . 7 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
2		bnj151.2	. . . . . . 7 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj151.6	. . . . . . 7 |- ( ze <-> ( ( R FrSe A /\ x e. A ) -> ( f Fn n /\ ph /\ ps ) ) )
4		bnj151.7	. . . . . . 7 |- ( ph' <-> [. 1o / n ]. ph )
5		bnj151.8	. . . . . . 7 |- ( ps' <-> [. 1o / n ]. ps )
6		bnj151.10	. . . . . . 7 |- ( th0 <-> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn 1o /\ ph' /\ ps' ) ) )
7		bnj151.12	. . . . . . 7 |- ( ze' <-> [. 1o / n ]. ze )
8		bnj151.13	. . . . . . 7 |- F = { <. (/) , pred ( x , A , R ) >. }
9		bnj151.14	. . . . . . 7 |- ( ph" <-> [. F / f ]. ph' )
10		bnj151.15	. . . . . . 7 |- ( ps" <-> [. F / f ]. ps' )
11		bnj151.16	. . . . . . 7 |- ( ze" <-> [. F / f ]. ze' )
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj150 30946	. . . . . 6 |- th0
13	12, 6	mpbi 220	. . . . 5 |- ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn 1o /\ ph' /\ ps' ) )
14		bnj151.11	. . . . . . 7 |- ( th1 <-> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) ) )
15		bnj151.17	. . . . . . 7 |- ( ze0 <-> ( f Fn 1o /\ ph' /\ ps' ) )
16		bnj151.18	. . . . . . 7 |- ( ze1 <-> [. g / f ]. ze0 )
17		bnj151.19	. . . . . . 7 |- ( ph1 <-> [. g / f ]. ph' )
18		bnj151.20	. . . . . . 7 |- ( ps1 <-> [. g / f ]. ps' )
19	1, 4	bnj118 30939	. . . . . . 7 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
20	14, 15, 16, 17, 18, 19	bnj149 30945	. . . . . 6 |- th1
21	20, 14	mpbi 220	. . . . 5 |- ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn 1o /\ ph' /\ ps' ) )
22		eu5 2496	. . . . 5 |- ( E! f ( f Fn 1o /\ ph' /\ ps' ) <-> ( E. f ( f Fn 1o /\ ph' /\ ps' ) /\ E* f ( f Fn 1o /\ ph' /\ ps' ) ) )
23	13, 21, 22	sylanbrc 698	. . . 4 |- ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) )
24		bnj151.4	. . . . 5 |- ( th <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
25		bnj151.9	. . . . 5 |- ( th' <-> [. 1o / n ]. th )
26	24, 4, 5, 25	bnj130 30944	. . . 4 |- ( th' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn 1o /\ ph' /\ ps' ) ) )
27	23, 26	mpbir 221	. . 3 |- th'
28		sbceq1a 3446	. . . 4 |- ( n = 1o -> ( th <-> [. 1o / n ]. th ) )
29	28, 25	syl6bbr 278	. . 3 |- ( n = 1o -> ( th <-> th' ) )
30	27, 29	mpbiri 248	. 2 |- ( n = 1o -> th )
31	30	a1d 25	1 |- ( n = 1o -> ( ( n e. D /\ ta ) -> th ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   A.wral 2912   [.wsbc 3435   \ cdif 3571   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   class class class wbr 4653   _E cep 5028   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   1oc1o 7553   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  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-1o 7560  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by:  bnj153  30950