Theorem bnj571 30976

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
bnj571.3	|- D = ( om \ { (/) } )
bnj571.16	|- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
bnj571.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj571.18	|- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
bnj571.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj571.20	|- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
bnj571.22	|- B = U_ y e. ( f ` i ) pred ( y , A , R )
bnj571.23	|- C = U_ y e. ( f ` p ) pred ( y , A , R )
bnj571.24	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj571.25	|- L = U_ y e. ( G ` p ) pred ( y , A , R )
bnj571.26	|- G = ( f u. { <. m , C >. } )
bnj571.29	|- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
bnj571.30	|- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj571.38	|- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
bnj571.21	|- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
bnj571.40	|- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
bnj571.33	|- ( ps" <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj571	|- ( ( R FrSe A /\ ta /\ et ) -> ps" )

Distinct variable groups:   A, i, p, y   y, G   R, i, p, y   et, i   f, i, p, y   i, m, p   i, ph', p

Allowed substitution hints:   ta( x, y, f, i, m, n, p)   et( x, y, f, 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, f, m, n)   B( x, y, f, i, m, n, p)   C( x, y, f, i, m, n, p)   D( x, y, f, i, m, n, p)   R( x, f, m, n)   G( x, f, i, 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)   ps"( x, y, f, i, m, n, p)

Proof of Theorem bnj571

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 |- F/ i R FrSe A
2		bnj571.17	. . . . 5 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
3		nfv 1843	. . . . . 6 |- F/ i f Fn m
4		nfv 1843	. . . . . 6 |- F/ i ph'
5		bnj571.30	. . . . . . 7 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
6		nfra1 2941	. . . . . . 7 |- F/ i A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
7	5, 6	nfxfr 1779	. . . . . 6 |- F/ i ps'
8	3, 4, 7	nf3an 1831	. . . . 5 |- F/ i ( f Fn m /\ ph' /\ ps' )
9	2, 8	nfxfr 1779	. . . 4 |- F/ i ta
10		nfv 1843	. . . 4 |- F/ i et
11	1, 9, 10	nf3an 1831	. . 3 |- F/ i ( R FrSe A /\ ta /\ et )
12		df-bnj17 30753	. . . . . . . . 9 |- ( ( R FrSe A /\ ta /\ et /\ ze ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ze ) )
13		3anass 1042	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) /\ m = suc i ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ( ( i e. om /\ suc i e. n ) /\ m = suc i ) ) )
14		3anrot 1043	. . . . . . . . . 10 |- ( ( m = suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) /\ m = suc i ) )
15		bnj571.20	. . . . . . . . . . . 12 |- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
16		df-3an 1039	. . . . . . . . . . . 12 |- ( ( i e. om /\ suc i e. n /\ m = suc i ) <-> ( ( i e. om /\ suc i e. n ) /\ m = suc i ) )
17	15, 16	bitri 264	. . . . . . . . . . 11 |- ( ze <-> ( ( i e. om /\ suc i e. n ) /\ m = suc i ) )
18	17	anbi2i 730	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ ta /\ et ) /\ ze ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ( ( i e. om /\ suc i e. n ) /\ m = suc i ) ) )
19	13, 14, 18	3bitr4ri 293	. . . . . . . . 9 |- ( ( ( R FrSe A /\ ta /\ et ) /\ ze ) <-> ( m = suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) )
20	12, 19	bitri 264	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et /\ ze ) <-> ( m = suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) )
21		bnj571.3	. . . . . . . . 9 |- D = ( om \ { (/) } )
22		bnj571.16	. . . . . . . . 9 |- G = ( f u. { <. m , U_ y e. ( f ` p ) pred ( y , A , R ) >. } )
23		bnj571.18	. . . . . . . . 9 |- ( si <-> ( m e. D /\ n = suc m /\ p e. m ) )
24		bnj571.19	. . . . . . . . 9 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
25		bnj571.22	. . . . . . . . 9 |- B = U_ y e. ( f ` i ) pred ( y , A , R )
26		bnj571.23	. . . . . . . . 9 |- C = U_ y e. ( f ` p ) pred ( y , A , R )
27		bnj571.24	. . . . . . . . 9 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
28		bnj571.25	. . . . . . . . 9 |- L = U_ y e. ( G ` p ) pred ( y , A , R )
29		bnj571.26	. . . . . . . . 9 |- G = ( f u. { <. m , C >. } )
30		bnj571.29	. . . . . . . . 9 |- ( ph' <-> ( f ` (/) ) = pred ( x , A , R ) )
31		bnj571.38	. . . . . . . . 9 |- ( ( R FrSe A /\ ta /\ si ) -> G Fn n )
32	21, 22, 2, 23, 24, 15, 25, 26, 27, 28, 29, 30, 5, 31	bnj558 30972	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et /\ ze ) -> ( G ` suc i ) = K )
33	20, 32	sylbir 225	. . . . . . 7 |- ( ( m = suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) -> ( G ` suc i ) = K )
34	33	3expib 1268	. . . . . 6 |- ( m = suc i -> ( ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) -> ( G ` suc i ) = K ) )
35		df-bnj17 30753	. . . . . . . . 9 |- ( ( R FrSe A /\ ta /\ et /\ rh ) <-> ( ( R FrSe A /\ ta /\ et ) /\ rh ) )
36		3anass 1042	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) /\ m =/= suc i ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ( ( i e. om /\ suc i e. n ) /\ m =/= suc i ) ) )
37		3anrot 1043	. . . . . . . . . 10 |- ( ( m =/= suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) /\ m =/= suc i ) )
38		bnj571.21	. . . . . . . . . . . 12 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
39		df-3an 1039	. . . . . . . . . . . 12 |- ( ( i e. om /\ suc i e. n /\ m =/= suc i ) <-> ( ( i e. om /\ suc i e. n ) /\ m =/= suc i ) )
40	38, 39	bitri 264	. . . . . . . . . . 11 |- ( rh <-> ( ( i e. om /\ suc i e. n ) /\ m =/= suc i ) )
41	40	anbi2i 730	. . . . . . . . . 10 |- ( ( ( R FrSe A /\ ta /\ et ) /\ rh ) <-> ( ( R FrSe A /\ ta /\ et ) /\ ( ( i e. om /\ suc i e. n ) /\ m =/= suc i ) ) )
42	36, 37, 41	3bitr4ri 293	. . . . . . . . 9 |- ( ( ( R FrSe A /\ ta /\ et ) /\ rh ) <-> ( m =/= suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) )
43	35, 42	bitri 264	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et /\ rh ) <-> ( m =/= suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) )
44		bnj571.40	. . . . . . . . 9 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
45	21, 2, 24, 38, 27, 22, 44, 5	bnj570 30975	. . . . . . . 8 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = K )
46	43, 45	sylbir 225	. . . . . . 7 |- ( ( m =/= suc i /\ ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) -> ( G ` suc i ) = K )
47	46	3expib 1268	. . . . . 6 |- ( m =/= suc i -> ( ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) -> ( G ` suc i ) = K ) )
48	34, 47	pm2.61ine 2877	. . . . 5 |- ( ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) -> ( G ` suc i ) = K )
49	48, 27	syl6eq 2672	. . . 4 |- ( ( ( R FrSe A /\ ta /\ et ) /\ ( i e. om /\ suc i e. n ) ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
50	49	exp32 631	. . 3 |- ( ( R FrSe A /\ ta /\ et ) -> ( i e. om -> ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) ) )
51	11, 50	alrimi 2082	. 2 |- ( ( R FrSe A /\ ta /\ et ) -> A. i ( i e. om -> ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) ) )
52		bnj571.33	. . 3 |- ( ps" <-> A. i e. om ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) )
53	52	bnj946 30845	. 2 |- ( ps" <-> A. i ( i e. om -> ( suc i e. n -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) ) ) )
54	51, 53	sylibr 224	1 |- ( ( R FrSe A /\ ta /\ et ) -> ps" )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ 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-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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  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-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-res 5126  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj600  30989  bnj908  31001