Theorem bnj908 29556

Description: Technical lemma for bnj69 29633. 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 29319	. . . . . 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 3081	. . . . . . . . . . 11 |- m e. _V
7	2, 3, 4, 5, 6	bnj207 29506	. . . . . . . . . 10 |- ( ch' <-> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
8	7	biimpi 197	. . . . . . . . 9 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E! f ( f Fn m /\ ph' /\ ps' ) ) )
9		euex 2288	. . . . . . . . 9 |- ( E! f ( f Fn m /\ ph' /\ ps' ) -> E. f ( f Fn m /\ ph' /\ ps' ) )
10	8, 9	syl6 34	. . . . . . . 8 |- ( ch' -> ( ( R FrSe A /\ x e. A ) -> E. f ( f Fn m /\ ph' /\ ps' ) ) )
11	10	impcom 431	. . . . . . 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 29421	. . . . . 6 |- ( ( ( R FrSe A /\ x e. A ) /\ ch' ) -> E. f ta )
14	1, 13	bnj832 29382	. . . . 5 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ta )
15		bnj645 29374	. . . . 5 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> et )
16		19.41v 1819	. . . . 5 |- ( E. f ( ta /\ et ) <-> ( E. f ta /\ et ) )
17	14, 15, 16	sylanbrc 668	. . . 4 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ta /\ et ) )
18		bnj642 29372	. . . 4 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> R FrSe A )
19		19.41v 1819	. . . 4 |- ( E. f ( ( ta /\ et ) /\ R FrSe A ) <-> ( E. f ( ta /\ et ) /\ R FrSe A ) )
20	17, 18, 19	sylanbrc 668	. . 3 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( ( ta /\ et ) /\ R FrSe A ) )
21		bnj170 29317	. . 3 |- ( ( R FrSe A /\ ta /\ et ) <-> ( ( ta /\ et ) /\ R FrSe A ) )
22	20, 21	bnj1198 29421	. 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 29512	. . . . 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 29516	. . . . 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 29519	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
32	23, 24, 31	bnj561 29528	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
33		bnj908.13	. . . . . 6 |- ( ph" <-> [. G / f ]. ph )
34	30	bnj528 29514	. . . . . 6 |- G e. _V
35	25, 33, 34	bnj609 29542	. . . . 5 |- ( ph" <-> ( G ` (/) ) = pred ( x , A , R ) )
36	26, 29, 30, 12, 23, 31, 35	bnj545 29520	. . . 4 |- ( ( R FrSe A /\ ta /\ si ) -> ph" )
37	23, 24, 36	bnj562 29529	. . 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 29543	. . . 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 29531	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )
48	32, 37, 47	3jca 1185	. 2 |- ( ( R FrSe A /\ ta /\ et ) -> ( G Fn n /\ ph" /\ ps" ) )
49	22, 48	bnj593 29369	1 |- ( ( R FrSe A /\ x e. A /\ ch' /\ et ) -> E. f ( G Fn n /\ ph" /\ ps" ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   E.wex 1659   e. wcel 1867   E!weu 2263   =/= wne 2616   A.wral 2773   [.wsbc 3296   \ cdif 3430   u. cun 3431   (/)c0 3758   {csn 3993   <.cop 3999   U_ciun 4293   class class class wbr 4417   _E cep 4754   suc csuc 5435   Fn wfn 5587   ` cfv 5592   omcom 6697   /\ w-bnj17 29305   predc-bnj14 29307   FrSe w-bnj15 29311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588  ax-reg 8098

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-om 6698  df-bnj17 29306  df-bnj14 29308  df-bnj13 29310  df-bnj15 29312

This theorem is referenced by: (None)