Theorem bnj967 31015

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
bnj967.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj967.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj967.10	|- D = ( om \ { (/) } )
bnj967.12	|- C = U_ y e. ( f ` m ) pred ( y , A , R )
bnj967.13	|- G = ( f u. { <. n , C >. } )
bnj967.44	|- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )

Assertion

Ref	Expression
bnj967	|- ( ( ( 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 ) )

Distinct variable groups:   y, f   y, i   y, n

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

Proof of Theorem bnj967

Step	Hyp	Ref	Expression
1		bnj967.44	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) ) -> C e. _V )
2	1	3adant3 1081	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> C e. _V )
3		bnj967.3	. . . . . . . . 9 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
4	3	bnj1235 30875	. . . . . . . 8 |- ( ch -> f Fn n )
5	4	3ad2ant1 1082	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> f Fn n )
6	5	3ad2ant2 1083	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> f Fn n )
7		simp23 1096	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> p = suc n )
8		simp3 1063	. . . . . . 7 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) -> suc i e. n )
9	8	3ad2ant3 1084	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> suc i e. n )
10	2, 6, 7, 9	bnj951 30846	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) )
11		bnj967.10	. . . . . . . . . 10 |- D = ( om \ { (/) } )
12	11	bnj923 30838	. . . . . . . . 9 |- ( n e. D -> n e. om )
13	3, 12	bnj769 30832	. . . . . . . 8 |- ( ch -> n e. om )
14	13	3ad2ant1 1082	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> n e. om )
15	14, 8	bnj240 30765	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( n e. om /\ suc i e. n ) )
16		nnord 7073	. . . . . . . 8 |- ( n e. om -> Ord n )
17		ordtr 5737	. . . . . . . 8 |- ( Ord n -> Tr n )
18	16, 17	syl 17	. . . . . . 7 |- ( n e. om -> Tr n )
19		trsuc 5810	. . . . . . 7 |- ( ( Tr n /\ suc i e. n ) -> i e. n )
20	18, 19	sylan 488	. . . . . 6 |- ( ( n e. om /\ suc i e. n ) -> i e. n )
21	15, 20	syl 17	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> i e. n )
22		bnj658 30821	. . . . . . 7 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) -> ( C e. _V /\ f Fn n /\ p = suc n ) )
23	22	anim1i 592	. . . . . 6 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) /\ i e. n ) -> ( ( C e. _V /\ f Fn n /\ p = suc n ) /\ i e. n ) )
24		df-bnj17 30753	. . . . . 6 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) <-> ( ( C e. _V /\ f Fn n /\ p = suc n ) /\ i e. n ) )
25	23, 24	sylibr 224	. . . . 5 |- ( ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) /\ i e. n ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
26	10, 21, 25	syl2anc 693	. . . 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 ) ) -> ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) )
27		bnj967.13	. . . . 5 |- G = ( f u. { <. n , C >. } )
28	27	bnj945 30844	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
29	26, 28	syl 17	. . 3 |- ( ( ( 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 ` i ) = ( f ` i ) )
30	27	bnj945 30844	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) -> ( G ` suc i ) = ( f ` suc i ) )
31	10, 30	syl 17	. . 3 |- ( ( ( 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 ) = ( f ` suc i ) )
32		3simpb 1059	. . . 4 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) -> ( i e. om /\ suc i e. n ) )
33	32	3ad2ant3 1084	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ( i e. om /\ suc i e. n ) )
34	3	bnj1254 30880	. . . . 5 |- ( ch -> ps )
35	34	3ad2ant1 1082	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ps )
36	35	3ad2ant2 1083	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ ( ch /\ n = suc m /\ p = suc n ) /\ ( i e. om /\ suc i e. p /\ suc i e. n ) ) -> ps )
37	29, 31, 33, 36	bnj951 30846	. 2 |- ( ( ( 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 ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) )
38		bnj967.2	. . 3 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
39		bnj967.12	. . . 4 |- C = U_ y e. ( f ` m ) pred ( y , A , R )
40	39, 27	bnj958 31010	. . 3 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )
41	38, 40	bnj953 31009	. 2 |- ( ( ( G ` i ) = ( f ` i ) /\ ( G ` suc i ) = ( f ` suc i ) /\ ( i e. om /\ suc i e. n ) /\ ps ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
42	37, 41	syl 17	1 |- ( ( ( 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 ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   Tr wtr 4752   Ord word 5722   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:  bnj910  31018