Theorem bnj908 31001

Description: Technical lemma for bnj69 31078. 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 30766	. . . . . 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 3203	. . . . . . . . . . 11 |- m e. _V
7	2, 3, 4, 5, 6	bnj207 30951	. . . . . . . . . 10 |- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
8	7	biimpi 206	. . . . . . . . 9 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
9		euex 2494	. . . . . . . . 9 |- ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
10	8, 9	syl6 35	. . . . . . . 8 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) )
11	10	impcom 446	. . . . . . 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 30866	. . . . . 6 |- ( ( ( R FrSe A /\ x e. A ) /\ ch' ) -> E. f ta )
14	1, 13	bnj832 30828	. . . . 5 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ta )
15		bnj645 30820	. . . . 5 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> et )
16		19.41v 1914	. . . . 5 |- ( E. f ( ta /\ et ) <-> ( E. f ta /\ et ) )
17	14, 15, 16	sylanbrc 698	. . . 4 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ta /\ et ) )
18		bnj642 30818	. . . 4 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> R FrSe A )
19		19.41v 1914	. . . 4 |- ( E. f ( ( ta /\ et ) /\ R FrSe A ) <-> ( E. f ( ta /\ et ) /\ R FrSe A ) )
20	17, 18, 19	sylanbrc 698	. . 3 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ( ta /\ et ) /\ R FrSe A ) )
21		bnj170 30764	. . 3 |- ( ( R FrSe A /\ ta /\ et ) <-> ( ( ta /\ et ) /\ R FrSe A ) )
22	20, 21	bnj1198 30866	. 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 30957	. . . . 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 30961	. . . . 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 30964	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
32	23, 24, 31	bnj561 30973	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
33		bnj908.13	. . . . . 6 |- ( ph" <-> [. G / f ]. ph )
34	30	bnj528 30959	. . . . . 6 |- G e. _V
35	25, 33, 34	bnj609 30987	. . . . 5 |- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
36	26, 29, 30, 12, 23, 31, 35	bnj545 30965	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )
37	23, 24, 36	bnj562 30974	. . 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 30988	. . . 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 30976	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )
48	32, 37, 47	3jca 1242	. 2 |- ( ( R FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) )
49	22, 48	bnj593 30815	1 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   =/= wne 2794   A.wral 2912   [.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   /\ 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-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-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-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759

This theorem is referenced by: (None)