Theorem bnj570 33259

Description: Technical lemma for bnj852 33275. 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
bnj570.3	|- D = ( om \ { (/) } )
bnj570.17	|- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
bnj570.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj570.21	|- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
bnj570.24	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj570.26	|- G = ( f u. { <. m , C >. } )
bnj570.40	|- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
bnj570.30	|- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj570	|- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = K )

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

Allowed substitution hints:   ta( y, f, i, m, n, p)   et( y, f, i, m, n, p)   rh( 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( f, i, m, n, p)   K( y, f, i, m, n, p)   ph'( y, f, i, m, n, p)   ps'( y, f, i, m, n, p)

Proof of Theorem bnj570

Step	Hyp	Ref	Expression
1		bnj251 33051	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) <-> ( R FrSe A /\ ( ta /\ ( et /\ rh ) ) ) )
2		bnj570.17	. . . . . 6 |- ( ta <-> ( f Fn m /\ ph' /\ ps' ) )
3	2	simp3bi 1013	. . . . 5 |- ( ta -> ps' )
4		bnj570.21	. . . . . . . 8 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
5	4	simp1bi 1011	. . . . . . 7 |- ( rh -> i e. om )
6	5	adantl 466	. . . . . 6 |- ( ( et /\ rh ) -> i e. om )
7		bnj570.19	. . . . . . 7 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
8	7, 4	bnj563 33096	. . . . . 6 |- ( ( et /\ rh ) -> suc i e. m )
9	6, 8	jca 532	. . . . 5 |- ( ( et /\ rh ) -> ( i e. om /\ suc i e. m ) )
10		bnj570.30	. . . . . . . 8 |- ( ps' <-> A. i e. om ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
11	10	bnj946 33129	. . . . . . 7 |- ( ps' <-> A. i ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
12		sp 1808	. . . . . . 7 |- ( A. i ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) -> ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
13	11, 12	sylbi 195	. . . . . 6 |- ( ps' -> ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
14	13	imp32 433	. . . . 5 |- ( ( ps' /\ ( i e. om /\ suc i e. m ) ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
15	3, 9, 14	syl2an 477	. . . 4 |- ( ( ta /\ ( et /\ rh ) ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
16	1, 15	bnj833 33112	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
17		bnj570.40	. . . . . 6 |- ( ( R FrSe A /\ ta /\ et ) -> G Fn n )
18	17	bnj930 33124	. . . . 5 |- ( ( R FrSe A /\ ta /\ et ) -> Fun G )
19	18	bnj721 33110	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> Fun G )
20		bnj570.26	. . . . . 6 |- G = ( f u. { <. m , C >. } )
21	20	bnj931 33125	. . . . 5 |- f C_ G
22	21	a1i 11	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> f C_ G )
23		bnj667 33105	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( ta /\ et /\ rh ) )
24	2	bnj564 33097	. . . . . . 7 |- ( ta -> dom f = m )
25		eleq2 2540	. . . . . . . 8 |- ( dom f = m -> ( suc i e. dom f <-> suc i e. m ) )
26	25	biimpar 485	. . . . . . 7 |- ( ( dom f = m /\ suc i e. m ) -> suc i e. dom f )
27	24, 8, 26	syl2an 477	. . . . . 6 |- ( ( ta /\ ( et /\ rh ) ) -> suc i e. dom f )
28	27	3impb 1192	. . . . 5 |- ( ( ta /\ et /\ rh ) -> suc i e. dom f )
29	23, 28	syl 16	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> suc i e. dom f )
30	19, 22, 29	bnj1502 33202	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = ( f ` suc i ) )
31	2	simp1bi 1011	. . . . . . . . 9 |- ( ta -> f Fn m )
32		bnj252 33052	. . . . . . . . . . . . . 14 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) <-> ( m e. D /\ ( n = suc m /\ p e. om /\ m = suc p ) ) )
33	32	simplbi 460	. . . . . . . . . . . . 13 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) -> m e. D )
34	7, 33	sylbi 195	. . . . . . . . . . . 12 |- ( et -> m e. D )
35		eldifi 3626	. . . . . . . . . . . . 13 |- ( m e. ( om \ { (/) } ) -> m e. om )
36		bnj570.3	. . . . . . . . . . . . 13 |- D = ( om \ { (/) } )
37	35, 36	eleq2s 2575	. . . . . . . . . . . 12 |- ( m e. D -> m e. om )
38		nnord 6693	. . . . . . . . . . . 12 |- ( m e. om -> Ord m )
39	34, 37, 38	3syl 20	. . . . . . . . . . 11 |- ( et -> Ord m )
40	39	adantr 465	. . . . . . . . . 10 |- ( ( et /\ rh ) -> Ord m )
41	40, 8	jca 532	. . . . . . . . 9 |- ( ( et /\ rh ) -> ( Ord m /\ suc i e. m ) )
42	31, 41	anim12i 566	. . . . . . . 8 |- ( ( ta /\ ( et /\ rh ) ) -> ( f Fn m /\ ( Ord m /\ suc i e. m ) ) )
43		fndm 5680	. . . . . . . . 9 |- ( f Fn m -> dom f = m )
44		elelsuc 4950	. . . . . . . . . 10 |- ( suc i e. m -> suc i e. suc m )
45		ordsucelsuc 6642	. . . . . . . . . . 11 |- ( Ord m -> ( i e. m <-> suc i e. suc m ) )
46	45	biimpar 485	. . . . . . . . . 10 |- ( ( Ord m /\ suc i e. suc m ) -> i e. m )
47	44, 46	sylan2 474	. . . . . . . . 9 |- ( ( Ord m /\ suc i e. m ) -> i e. m )
48	43, 47	anim12i 566	. . . . . . . 8 |- ( ( f Fn m /\ ( Ord m /\ suc i e. m ) ) -> ( dom f = m /\ i e. m ) )
49		eleq2 2540	. . . . . . . . 9 |- ( dom f = m -> ( i e. dom f <-> i e. m ) )
50	49	biimpar 485	. . . . . . . 8 |- ( ( dom f = m /\ i e. m ) -> i e. dom f )
51	42, 48, 50	3syl 20	. . . . . . 7 |- ( ( ta /\ ( et /\ rh ) ) -> i e. dom f )
52	51	3impb 1192	. . . . . 6 |- ( ( ta /\ et /\ rh ) -> i e. dom f )
53	23, 52	syl 16	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> i e. dom f )
54	19, 22, 53	bnj1502 33202	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` i ) = ( f ` i ) )
55	54	iuneq1d 4350	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> U_ y e. ( G ` i ) pred ( y , A , R ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
56	16, 30, 55	3eqtr4d 2518	. 2 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
57		bnj570.24	. 2 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
58	56, 57	syl6eqr 2526	1 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = K )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   A.wal 1377   = wceq 1379   e. wcel 1767   =/= wne 2662   A.wral 2814   \ cdif 3473   u. cun 3474   C_ wss 3476   (/)c0 3785   {csn 4027   <.cop 4033   U_ciun 4325   Ord word 4877   suc csuc 4880   dom cdm 4999   Fun wfun 5582   Fn wfn 5583   ` cfv 5588   omcom 6685   /\ w-bnj17 33035   predc-bnj14 33037   FrSe w-bnj15 33041

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 6577

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 6686  df-bnj17 33036

This theorem is referenced by:  bnj571  33260