Theorem bnj908 33285

Description: Technical lemma for bnj69 33362. 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
bnj908.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj908.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj908.3	|- D = ( om \ { (/) } )
bnj908.4	|- ( ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj908.5	|- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
bnj908.10	|- ( ph' <-> [. m / n ]. ph )
bnj908.11	|- ( ps' <-> [. m / n ]. ps )
bnj908.12	|- ( ch' <-> [. m / n ]. ch )
bnj908.13	|- ( ph" <-> [. G / f ]. ph )
bnj908.14	|- ( ps" <-> [. G / f ]. ps )
bnj908.15	|- ( ch" <-> [. G / f ]. ch )
bnj908.16	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj908.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj908.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj908.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj908.20	|- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
bnj908.21	|- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
bnj908.22	|- B = U_ y e. ( f ` i ) pred ( y , A , R )
bnj908.23	|- C = U_ y e. ( f ` p ) pred ( y , A , R )
bnj908.24	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj908.25	|- L = U_ y e. ( G ` p ) pred ( y , A , R )
bnj908.26	|- G = ( f u. { <. m , C >. } )

Assertion

Ref	Expression
bnj908	|- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )

Distinct variable groups:   A, f, i, m, n, p   y, A, f, i, n, p   D, p   i, G, y   R, f, i, m, n, p   y, R   et, f, i   x, f, m, n, p   i, ph', p   ph, m, p   ps, m, p   th, p

Allowed substitution hints:   ph( x, y, f, i, n)   ps( x, y, f, i, n)   ch( x, y, f, i, m, n, p)   th( x, y, f, i, m, n)   ta( x, y, f, i, m, n, p)   et( x, y, m, n, p)   ze( x, y, f, i, m, n, p)   si( x, y, f, i, m, n, p)   rh( x, y, f, i, m, n, p)   A( x)   B( x, y, f, i, m, n, p)   C( x, y, f, i, m, n, p)   D( x, y, f, i, m, n)   R( x)   G( x, f, m, n, p)   K( x, y, f, i, m, n, p)   L( x, y, f, i, m, n, p)   ph'( x, y, f, m, n)   ps'( x, y, f, i, m, n, p)   ch'( x, y, f, i, m, n, p)   ph"( x, y, f, i, m, n, p)   ps"( x, y, f, i, m, n, p)   ch"( x, y, f, i, m, n, p)

Proof of Theorem bnj908

Step	Hyp	Ref	Expression
1		bnj248 33049	. . . . . 6 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) <-> ( ( ( R FrSe A /\ x e. A ) /\ ch' ) /\ et ) )
2		bnj908.4	. . . . . . . . . . 11 |- ( ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
3		bnj908.10	. . . . . . . . . . 11 |- ( ph' <-> [. m / n ]. ph )
4		bnj908.11	. . . . . . . . . . 11 |- ( ps' <-> [. m / n ]. ps )
5		bnj908.12	. . . . . . . . . . 11 |- ( ch' <-> [. m / n ]. ch )
6		vex 3116	. . . . . . . . . . 11 |- m e. _V
7	2, 3, 4, 5, 6	bnj207 33235	. . . . . . . . . 10 |- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
8	7	biimpi 194	. . . . . . . . 9 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
9		euex 2303	. . . . . . . . 9 |- ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
10	8, 9	syl6 33	. . . . . . . 8 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) )
11	10	impcom 430	. . . . . . 7 |- ( ( ( R FrSe A /\ x e. A ) /\ ch' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
12		bnj908.17	. . . . . . 7 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
13	11, 12	bnj1198 33150	. . . . . 6 |- ( ( ( R FrSe A /\ x e. A ) /\ ch' ) -> E. f ta )
14	1, 13	bnj832 33111	. . . . 5 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ta )
15		bnj645 33103	. . . . 5 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> et )
16		19.41v 1945	. . . . 5 |- ( E. f ( ta /\ et ) <-> ( E. f ta /\ et ) )
17	14, 15, 16	sylanbrc 664	. . . 4 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ta /\ et ) )
18		bnj642 33101	. . . 4 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> R FrSe A )
19		19.41v 1945	. . . 4 |- ( E. f ( ( ta /\ et ) /\ R FrSe A ) <-> ( E. f ( ta /\ et ) /\ R FrSe A ) )
20	17, 18, 19	sylanbrc 664	. . 3 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ( ta /\ et ) /\ R FrSe A ) )
21		bnj170 33047	. . 3 |- ( ( R FrSe A /\ ta /\ et ) <-> ( ( ta /\ et ) /\ R FrSe A ) )
22	20, 21	bnj1198 33150	. 2 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( R FrSe A /\ ta /\ et ) )
23		bnj908.18	. . . 4 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
24		bnj908.19	. . . 4 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
25		bnj908.1	. . . . . 6 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
26	25, 3, 6	bnj523 33241	. . . . 5 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
27		bnj908.2	. . . . . 6 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
28	27, 4, 6	bnj539 33245	. . . . 5 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
29		bnj908.3	. . . . 5 |- D = ( om \ { (/) } )
30		bnj908.16	. . . . 5 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
31	26, 28, 29, 30, 12, 23	bnj544 33248	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
32	23, 24, 31	bnj561 33257	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
33		bnj908.13	. . . . . 6 |- ( ph" <-> [. G / f ]. ph )
34	30	bnj528 33243	. . . . . 6 |- G e. _V
35	25, 33, 34	bnj609 33271	. . . . 5 |- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
36	26, 29, 30, 12, 23, 31, 35	bnj545 33249	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )
37	23, 24, 36	bnj562 33258	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> ph" )
38		bnj908.20	. . . 4 |- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
39		bnj908.22	. . . 4 |- B = U_ y e. ( f ` i ) pred ( y , A , R )
40		bnj908.23	. . . 4 |- C = U_ y e. ( f ` p ) pred ( y , A , R )
41		bnj908.24	. . . 4 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
42		bnj908.25	. . . 4 |- L = U_ y e. ( G ` p ) pred ( y , A , R )
43		bnj908.26	. . . 4 |- G = ( f u. { <. m , C >. } )
44		bnj908.21	. . . 4 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
45		bnj908.14	. . . . 5 |- ( ps" <-> [. G / f ]. ps )
46	27, 45, 34	bnj611 33272	. . . 4 |- ( ps" <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
47	29, 30, 12, 23, 24, 38, 39, 40, 41, 42, 43, 26, 28, 31, 44, 32, 46	bnj571 33260	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )
48	32, 37, 47	3jca 1176	. 2 |- ( ( R FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) )
49	22, 48	bnj593 33098	1 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   E!weu 2275   =/= wne 2662   A.wral 2814   [.wsbc 3331   \ cdif 3473   u. cun 3474   (/)c0 3785   {csn 4027   <.cop 4033   U_ciun 4325   class class class wbr 4447   _E cep 4789   suc csuc 4880   Fn wfn 5583   ` cfv 5588   omcom 6685   /\ w-bnj17 33035   predc-bnj14 33037   FrSe w-bnj15 33041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577  ax-reg 8019

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-om 6686  df-bnj17 33036  df-bnj14 33038  df-bnj13 33040  df-bnj15 33042

This theorem is referenced by: (None)