Theorem bnj600 30989

Description: Technical lemma for bnj852 30991. 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 30959	. . . . . 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 3203	. . . . . . 7 |- m e. _V
13	8, 9, 10, 11, 12	bnj207 30951	. . . . . 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 30987	. . . . . 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 30988	. . . . . 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 30798	. . . . . . . . 9 |- ( ( n =/= 1o /\ n e. D ) -> E. m e. D n = suc m )
20		df-rex 2918	. . . . . . . . 9 |- ( E. m e. D n = suc m <-> E. m ( m e. D /\ n = suc m ) )
21	19, 20	sylib 208	. . . . . . . 8 |- ( ( n =/= 1o /\ n e. D ) -> E. m ( m e. D /\ n = suc m ) )
22	18	bnj158 30797	. . . . . . . . . . . . . 14 |- ( m e. D -> E. p e. om m = suc p )
23		df-rex 2918	. . . . . . . . . . . . . 14 |- ( E. p e. om m = suc p <-> E. p ( p e. om /\ m = suc p ) )
24	22, 23	sylib 208	. . . . . . . . . . . . 13 |- ( m e. D -> E. p ( p e. om /\ m = suc p ) )
25	24	adantr 481	. . . . . . . . . . . 12 |- ( ( m e. D /\ n = suc m ) -> E. p ( p e. om /\ m = suc p ) )
26	25	ancri 575	. . . . . . . . . . 11 |- ( ( m e. D /\ n = suc m ) -> ( E. p ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
27	26	bnj534 30808	. . . . . . . . . 10 |- ( ( m e. D /\ n = suc m ) -> E. p ( ( p e. om /\ m = suc p ) /\ ( m e. D /\ n = suc m ) ) )
28		bnj432 30782	. . . . . . . . . . 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 1774	. . . . . . . . . 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 224	. . . . . . . . 9 |- ( ( m e. D /\ n = suc m ) -> E. p ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
31	30	eximi 1762	. . . . . . . 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 1775	. . . . . . 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 224	. . . . . 6 |- ( ( n =/= 1o /\ n e. D ) -> E. m E. p et )
35		rsp 2929	. . . . . . . . 9 |- ( A. m e. D ( m _E n -> [. m / n ]. ch ) -> ( m e. D -> ( m _E n -> [. m / n ]. ch ) ) )
36	1, 35	sylbi 207	. . . . . . . 8 |- ( th -> ( m e. D -> ( m _E n -> [. m / n ]. ch ) ) )
37	36	3imp 1256	. . . . . . 7 |- ( ( th /\ m e. D /\ m _E n ) -> [. m / n ]. ch )
38	37, 11	sylibr 224	. . . . . 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 30957	. . . . . . . 8 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
41	16, 10, 12	bnj539 30961	. . . . . . . 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 30964	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
43	39, 5, 42	bnj561 30973	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
44	40, 18, 6, 4, 39, 42, 15	bnj545 30965	. . . . . . 7 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )
45	39, 5, 44	bnj562 30974	. . . . . 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 30976	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )
54		biid 251	. . . . . 6 |- ( [. z / f ]. ph <-> [. z / f ]. ph )
55		biid 251	. . . . . 6 |- ( [. z / f ]. ps <-> [. z / f ]. ps )
56		biid 251	. . . . . 6 |- ( [. G / z ]. [. z / f ]. ph <-> [. G / z ]. [. z / f ]. ph )
57		biid 251	. . . . . 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 30986	. . . . 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 30984	. . . . . . 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 1083	. . . . 5 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E* f ( f Fn n /\ ph /\ ps ) ) )
62	58, 61	jcad 555	. . . 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 2496	. . . 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 242	. . 3 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn n /\ ph /\ ps ) ) )
65	64, 8	sylibr 224	. 2 |- ( ( n =/= 1o /\ n e. D /\ th ) -> ch )
66	65	3expib 1268	1 |- ( n =/= 1o -> ( ( n e. D /\ th ) -> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   =/= wne 2794   A.wral 2912   E.wrex 2913   [.wsbc 3435   \ cdif 3571   u. cun 3572   (/)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   /\ w-bnj17 30752   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

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:  bnj601  30990