Theorem bnj910 31018

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
bnj910.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj910.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj910.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj910.4	|- ( ph' <-> [. p / n ]. ph )
bnj910.5	|- ( ps' <-> [. p / n ]. ps )
bnj910.6	|- ( ch' <-> [. p / n ]. ch )
bnj910.7	|- ( ph" <-> [. G / f ]. ph' )
bnj910.8	|- ( ps" <-> [. G / f ]. ps' )
bnj910.9	|- ( ch" <-> [. G / f ]. ch' )
bnj910.10	|- D = ( om \ { (/) } )
bnj910.11	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj910.12	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj910.13	|- G = ( f u. { <. n , C >. } )
bnj910.14	|- ( ta <-> ( f Fn n /\ ph /\ ps ) )
bnj910.15	|- ( si <-> ( n e. D /\ p = suc n /\ m e. n ) )

Assertion

Ref	Expression
bnj910	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ch" )

Distinct variable groups:   A, f, i, m, n, y   D, f, i, n   i, G   R, f, i, m, n, y   f, X, i, n   f, p, i, n   ph, i

Allowed substitution hints:   ph( y, f, m, n, p)   ps( y, f, i, m, n, p)   ch( y, f, i, m, n, p)   ta( y, f, i, m, n, p)   si( y, f, i, m, n, p)   A( p)   B( y, f, i, m, n, p)   C( y, f, i, m, n, p)   D( y, m, p)   R( p)   G( y, f, m, n, p)   X( y, m, p)   ph'( y, f, i, m, n, p)   ps'( y, f, i, m, n, p)   ch'( y, f, i, m, n, p)   ph"( y, f, i, m, n, p)   ps"( y, f, i, m, n, p)   ch"( y, f, i, m, n, p)

Proof of Theorem bnj910

Step	Hyp	Ref	Expression
1		bnj910.3	. . . 4 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj910.10	. . . 4 |- D = ( om \ { (/) } )
3	1, 2	bnj970 31017	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p e. D )
4		bnj910.1	. . . . 5 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
5		bnj910.2	. . . . 5 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
6		bnj910.12	. . . . 5 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
7		bnj910.14	. . . . 5 |- ( ta <-> ( f Fn n /\ ph /\ ps ) )
8		bnj910.15	. . . . 5 |- ( si <-> ( n e. D /\ p = suc n /\ m e. n ) )
9	4, 5, 1, 2, 6, 7, 8	bnj969 31016	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
10		simpr3 1069	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> p = suc n )
11	1	bnj1235 30875	. . . . . 6 |- ( ch -> f Fn n )
12	11	3ad2ant1 1082	. . . . 5 |- ( ( ch /\ n = suc m /\ p = suc n ) -> f Fn n )
13	12	adantl 482	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> f Fn n )
14		bnj910.13	. . . . . 6 |- G = ( f u. { <. n , C >. } )
15	14	bnj941 30843	. . . . 5 |- ( C e. _V -> ( ( p = suc n /\ f Fn n ) -> G Fn p ) )
16	15	3impib 1262	. . . 4 |- ( ( C e. _V /\ p = suc n /\ f Fn n ) -> G Fn p )
17	9, 10, 13, 16	syl3anc 1326	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> G Fn p )
18		bnj910.4	. . . 4 |- ( ph' <-> [. p / n ]. ph )
19		bnj910.7	. . . 4 |- ( ph" <-> [. G / f ]. ph' )
20	4, 5, 1, 18, 19, 2, 6, 14, 7, 8	bnj944 31008	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ph" )
21		bnj910.5	. . . 4 |- ( ps' <-> [. p / n ]. ps )
22		bnj910.8	. . . 4 |- ( ps" <-> [. G / f ]. ps' )
23	5, 1, 2, 6, 14, 9	bnj967 31015	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
24	1, 2, 6, 14, 9, 17	bnj966 31014	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ n = suc i ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
25	5, 1, 21, 22, 6, 14, 23, 24	bnj964 31013	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ps" )
26	3, 17, 20, 25	bnj951 30846	. 2 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
27		bnj910.6	. . . 4 |- ( ch' <-> [. p / n ]. ch )
28		vex 3203	. . . 4 |- p e. _V
29	1, 18, 21, 27, 28	bnj919 30837	. . 3 |- ( ch' <-> ( p e. D /\ f Fn p /\ ph' /\ ps' ) )
30		bnj910.9	. . 3 |- ( ch" <-> [. G / f ]. ch' )
31	14	bnj918 30836	. . 3 |- G e. _V
32	29, 19, 22, 30, 31	bnj976 30848	. 2 |- ( ch" <-> ( p e. D /\ G Fn p /\ ph" /\ ps" ) )
33	26, 32	sylibr 224	1 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> ch" )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   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:  bnj998  31026