Theorem bnj600 29291

Description: Technical lemma for bnj852 29293. 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
bnj600.1	|- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj600.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj600.3	|- D = ( om \ { (/) } )
bnj600.4	|- ( ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
bnj600.5	|- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
bnj600.10	|- ( ph' <-> [. m / n ]. ph )
bnj600.11	|- ( ps' <-> [. m / n ]. ps )
bnj600.12	|- ( ch' <-> [. m / n ]. ch )
bnj600.13	|- ( ph" <-> [. G / f ]. ph )
bnj600.14	|- ( ps" <-> [. G / f ]. ps )
bnj600.15	|- ( ch" <-> [. G / f ]. ch )
bnj600.16	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj600.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj600.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj600.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj600.20	|- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
bnj600.21	|- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
bnj600.22	|- B = U_ y e. ( f ` i ) pred ( y , A , R )
bnj600.23	|- C = U_ y e. ( f ` p ) pred ( y , A , R )
bnj600.24	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj600.25	|- L = U_ y e. ( G ` p ) pred ( y , A , R )
bnj600.26	|- G = ( f u. { <. m , C >. } )

Assertion

Ref	Expression
bnj600	|- ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )

Distinct variable groups:   A, f, i, m, n, p   y, A, f, i, n, p   D, f, 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, 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 bnj600

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj600.5	. . . . . 6 |- ( th <-> A. m e. D ( m _E n -> [. m / n ]. ch ) )
2		bnj600.13	. . . . . 6 |- ( ph" <-> [. G / f ]. ph )
3		bnj600.14	. . . . . 6 |- ( ps" <-> [. G / f ]. ps )
4		bnj600.17	. . . . . 6 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
5		bnj600.19	. . . . . 6 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
6		bnj600.16	. . . . . . 7 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
7	6	bnj528 29261	. . . . . 6 |- G e. _V
8		bnj600.4	. . . . . . 7 |- ( ch <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
9		bnj600.10	. . . . . . 7 |- ( ph' <-> [. m / n ]. ph )
10		bnj600.11	. . . . . . 7 |- ( ps' <-> [. m / n ]. ps )
11		bnj600.12	. . . . . . 7 |- ( ch' <-> [. m / n ]. ch )
12		vex 3061	. . . . . . 7 |- m e. _V
13	8, 9, 10, 11, 12	bnj207 29253	. . . . . 6 |- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
14		bnj600.1	. . . . . . 7 |- ( ph <-> ( f ` (/) ) = pred ( x , A , R ) )
15	14, 2, 7	bnj609 29289	. . . . . 6 |- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
16		bnj600.2	. . . . . . 7 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
17	16, 3, 7	bnj611 29290	. . . . . 6 |- ( ps" <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
18		bnj600.3	. . . . . . . . . 10 |- D = ( om \ { (/) } )
19	18	bnj168 29099	. . . . . . . . 9 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )
20		df-rex 2759	. . . . . . . . 9 |- ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
21	19, 20	sylib 196	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. D /\ n = suc m ) )
22	18	bnj158 29098	. . . . . . . . . . . . . 14 |- ( m e. D -> E. p e. om m = suc p )
23		df-rex 2759	. . . . . . . . . . . . . 14 |- ( E. p e. om m = suc p <-> E. p ( p e. om /\ m = suc p ) )
24	22, 23	sylib 196	. . . . . . . . . . . . 13 |- ( m e. D -> E. p ( p e. om /\ m = suc p ) )
25	24	adantr 463	. . . . . . . . . . . 12 |- ( ( m e. D /\ n = suc m ) -> E. p ( p e. om /\ m = suc p ) )
26	25	ancri 550	. . . . . . . . . . 11 |- ( ( m e. D /\ n = suc m ) -> ( E. p ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
27	26	bnj534 29109	. . . . . . . . . 10 |- ( ( m e. D /\ n = suc m ) -> E. p ( ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
28		bnj432 29082	. . . . . . . . . . 11 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) <-> ( ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
29	28	exbii 1688	. . . . . . . . . 10 |- ( E. p ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) <-> E. p ( ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
30	27, 29	sylibr 212	. . . . . . . . 9 |- ( ( m e. D /\ n = suc m ) -> E. p ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
31	30	eximi 1677	. . . . . . . 8 |- ( E. m ( m e. D /\ n = suc m ) -> E. m E. p ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
32	21, 31	syl 17	. . . . . . 7 |- ( ( n =/= 1o /\ n e. D ) -> E. m E. p ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
33	5	2exbii 1689	. . . . . . 7 |- ( E. m E. p et <-> E. m E. p ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
34	32, 33	sylibr 212	. . . . . 6 |- ( ( n =/= 1o /\ n e. D ) -> E. m E. p et )
35		rsp 2769	. . . . . . . . 9 |- ( A. m e. D ( m _E n -> [. m / n ]. ch ) -> ( m e. D -> ( m _E n -> [. m / n ]. ch ) ) )
36	1, 35	sylbi 195	. . . . . . . 8 |- ( th -> ( m e. D -> ( m _E n -> [. m / n ]. ch ) ) )
37	36	3imp 1191	. . . . . . 7 |- ( ( th /\ m e. D /\ m _E n ) -> [. m / n ]. ch )
38	37, 11	sylibr 212	. . . . . 6 |- ( ( th /\ m e. D /\ m _E n ) -> ch' )
39		bnj600.18	. . . . . . 7 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
40	14, 9, 12	bnj523 29259	. . . . . . . 8 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
41	16, 10, 12	bnj539 29263	. . . . . . . 8 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
42	40, 41, 18, 6, 4, 39	bnj544 29266	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
43	39, 5, 42	bnj561 29275	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
44	40, 18, 6, 4, 39, 42, 15	bnj545 29267	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )
45	39, 5, 44	bnj562 29276	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> ph" )
46		bnj600.20	. . . . . . 7 |- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
47		bnj600.22	. . . . . . 7 |- B = U_ y e. ( f ` i ) pred ( y , A , R )
48		bnj600.23	. . . . . . 7 |- C = U_ y e. ( f ` p ) pred ( y , A , R )
49		bnj600.24	. . . . . . 7 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
50		bnj600.25	. . . . . . 7 |- L = U_ y e. ( G ` p ) pred ( y , A , R )
51		bnj600.26	. . . . . . 7 |- G = ( f u. { <. m , C >. } )
52		bnj600.21	. . . . . . 7 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
53	18, 6, 4, 39, 5, 46, 47, 48, 49, 50, 51, 40, 41, 42, 52, 43, 17	bnj571 29278	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )
54		biid 236	. . . . . 6 |- ( [. z / f ]. ph <-> [. z / f ]. ph )
55		biid 236	. . . . . 6 |- ( [. z / f ]. ps <-> [. z / f ]. ps )
56		biid 236	. . . . . 6 |- ( [. G / z ]. [. z / f ]. ph <-> [. G / z ]. [. z / f ]. ph )
57		biid 236	. . . . . 6 |- ( [. G / z ]. [. z / f ]. ps <-> [. G / z ]. [. z / f ]. ps )
58	1, 2, 3, 4, 5, 7, 13, 15, 17, 34, 38, 43, 45, 53, 14, 16, 54, 55, 56, 57	bnj607 29288	. . . . 5 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn n /\ ph /\ ps ) ) )
59	14, 16, 18	bnj579 29286	. . . . . . 7 |- ( n e. D -> E* f ( f Fn n /\ ph /\ ps ) )
60	59	a1d 25	. . . . . 6 |- ( n e. D -> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn n /\ ph /\ ps ) ) )
61	60	3ad2ant2 1019	. . . . 5 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn n /\ ph /\ ps ) ) )
62	58, 61	jcad 531	. . . 4 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> ( E. f ( f Fn n /\ ph /\ ps ) /\ E* f ( f Fn n /\ ph /\ ps ) ) ) )
63		eu5 2266	. . . 4 |- ( E! f ( f Fn n /\ ph /\ ps ) <-> ( E. f ( f Fn n /\ ph /\ ps ) /\ E* f ( f Fn n /\ ph /\ ps ) ) )
64	62, 63	syl6ibr 227	. . 3 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
65	64, 8	sylibr 212	. 2 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ch )
66	65	3expib 1200	1 |- ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   E.wex 1633   e. wcel 1842   E!weu 2238   E*wmo 2239   =/= wne 2598   A.wral 2753   E.wrex 2754   [.wsbc 3276   \ cdif 3410   u. cun 3411   (/)c0 3737   {csn 3971   <.cop 3977   U_ciun 4270   class class class wbr 4394   _E cep 4731   suc csuc 5411   Fn wfn 5563   ` cfv 5568   omcom 6682   1oc1o 7159   /\ w-bnj17 29052   predc-bnj14 29054   FrSe w-bnj15 29058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-reg 8051

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-om 6683  df-1o 7166  df-bnj17 29053  df-bnj14 29055  df-bnj13 29057  df-bnj15 29059

This theorem is referenced by:  bnj601  29292