Theorem bnj600 32008

Description: Technical lemma for bnj852 32010. 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 31978	. . . . . 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 2996	. . . . . . 7 |- m e. _V
13	8, 9, 10, 11, 12	bnj207 31970	. . . . . 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 32006	. . . . . 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 32007	. . . . . 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 31817	. . . . . . . . 9 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )
20		df-rex 2742	. . . . . . . . 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 31816	. . . . . . . . . . . . . 14 |- ( m e. D -> E. p e. om m = suc p )
23		df-rex 2742	. . . . . . . . . . . . . 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 465	. . . . . . . . . . . 12 |- ( ( m e. D /\ n = suc m ) -> E. p ( p e. om /\ m = suc p ) )
26	25	ancri 552	. . . . . . . . . . 11 |- ( ( m e. D /\ n = suc m ) -> ( E. p ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
27	26	bnj534 31827	. . . . . . . . . 10 |- ( ( m e. D /\ n = suc m ) -> E. p ( ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
28		bnj432 31800	. . . . . . . . . . 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 1634	. . . . . . . . . 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 1625	. . . . . . . 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 16	. . . . . . 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 1635	. . . . . . 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 2797	. . . . . . . . 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 1181	. . . . . . 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 31976	. . . . . . . 8 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
41	16, 10, 12	bnj539 31980	. . . . . . . 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 31983	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
43	39, 5, 42	bnj561 31992	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
44	40, 18, 6, 4, 39, 42, 15	bnj545 31984	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )
45	39, 5, 44	bnj562 31993	. . . . . 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 31995	. . . . . 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 32005	. . . . 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 32003	. . . . . . 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 1010	. . . . 5 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn n /\ ph /\ ps ) ) )
62	58, 61	jcad 533	. . . 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 2283	. . . 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 1190	1 |- ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   E.wex 1586   e. wcel 1756   E!weu 2253   E*wmo 2254   =/= wne 2620   A.wral 2736   E.wrex 2737   [.wsbc 3207   \ cdif 3346   u. cun 3347   (/)c0 3658   {csn 3898   <.cop 3904   U_ciun 4192   class class class wbr 4313   _E cep 4651   suc csuc 4742   Fn wfn 5434   ` cfv 5439   omcom 6497   1oc1o 6934   /\ w-bnj17 31770   predc-bnj14 31772   FrSe w-bnj15 31776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-reg 7828

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-om 6498  df-1o 6941  df-bnj17 31771  df-bnj14 31773  df-bnj13 31775  df-bnj15 31777

This theorem is referenced by:  bnj601  32009