Theorem bnj967 33100

Description: Technical lemma for bnj69 33163. 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 32960	. . . . . . . 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 32931	. . . . 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 32923	. . . . . . . . 9 |- ( n e. D -> n e. om )
13	3, 12	bnj769 32917	. . . . . . . 8 |- ( ch -> n e. om )
14	13	3ad2ant1 1017	. . . . . . 7 |- ( ( ch /\ n = suc m /\ p = suc n ) -> n e. om )
15	14, 8	bnj240 32849	. . . . . 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 6692	. . . . . . . 8 |- ( n e. om -> Ord n )
17		ordtr 4892	. . . . . . . 8 |- ( Ord n -> Tr n )
18	16, 17	syl 16	. . . . . . 7 |- ( n e. om -> Tr n )
19		trsuc 4962	. . . . . . 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 32905	. . . . . . 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 32837	. . . . . 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 32929	. . . 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 32929	. . . 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 32965	. . . . 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 32931	. 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 33095	. . 3 |- ( ( G ` i ) = ( f ` i ) -> A. y ( G ` i ) = ( f ` i ) )
41	38, 40	bnj953 33094	. 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 1379   e. wcel 1767   A.wral 2814   _Vcvv 3113   \ cdif 3473   u. cun 3474   (/)c0 3785   {csn 4027   <.cop 4033   U_ciun 4325   Tr wtr 4540   Ord word 4877   suc csuc 4880   Fn wfn 5583   ` cfv 5588   omcom 6684   /\ w-bnj17 32836   predc-bnj14 32838   FrSe w-bnj15 32842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576  ax-reg 8018

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-om 6685  df-bnj17 32837

This theorem is referenced by:  bnj910  33103