Theorem bnj967 34146

Description: Technical lemma for bnj69 34209. 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 1016	. . . . . 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 34006	. . . . . . . 8 |- ( ch -> f Fn n )
5	4	3ad2ant1 1017	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> f Fn n )
6	5	3ad2ant2 1018	. . . . . 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 1031	. . . . . 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 998	. . . . . . 7 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) -> suc i e. n )
9	8	3ad2ant3 1019	. . . . . 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 33977	. . . . 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 33969	. . . . . . . . 9 |- ( n e. D -> n e. om )
13	3, 12	bnj769 33963	. . . . . . . 8 |- ( ch -> n e. om )
14	13	3ad2ant1 1017	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> n e. om )
15	14, 8	bnj240 33894	. . . . . 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 6707	. . . . . . . 8 |- ( n e. om -> Ord n )
17		ordtr 4901	. . . . . . . 8 |- ( Ord n -> Tr n )
18	16, 17	syl 16	. . . . . . 7 |- ( n e. om -> Tr n )
19		trsuc 4971	. . . . . . 7 |- ( ( Tr n /\ suc i e. n ) -> i e. n )
20	18, 19	sylan 471	. . . . . 6 |- ( ( n e. om /\ suc i e. n ) -> i e. n )
21	15, 20	syl 16	. . . . 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 33951	. . . . . . 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 568	. . . . . 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 33882	. . . . . 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 212	. . . . 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 661	. . . 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 33975	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ i e. n ) -> ( G ` i ) = ( f ` i ) )
29	26, 28	syl 16	. . 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 33975	. . . 4 |- ( ( C e. _V /\ f Fn n /\ p = suc n /\ suc i e. n ) -> ( G ` suc i ) = ( f ` suc i ) )
31	10, 30	syl 16	. . 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 994	. . . 4 |- ( ( i e. om /\ suc i e. p /\ suc i e. n ) -> ( i e. om /\ suc i e. n ) )
33	32	3ad2ant3 1019	. . 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 34011	. . . . 5 |- ( ch -> ps )
35	34	3ad2ant1 1017	. . . 4 |- ( ( ch /\ n = suc m /\ p = suc n ) -> ps )
36	35	3ad2ant2 1018	. . 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 33977	. 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 34141	. . 3 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )
41	38, 40	bnj953 34140	. 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 16	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 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   A.wral 2807   _Vcvv 3109   \ cdif 3468   u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038   U_ciun 4332   Tr wtr 4550   Ord word 4886   suc csuc 4889   Fn wfn 5589   ` cfv 5594   omcom 6699   /\ w-bnj17 33881   predc-bnj14 33883   FrSe w-bnj15 33887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591  ax-reg 8036

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-om 6700  df-bnj17 33882

This theorem is referenced by:  bnj910  34149