Theorem bnj570 30975

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
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 30768	. . . 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 1078	. . . . 5 |- ( ta -> ps' )
4		bnj570.21	. . . . . . . 8 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
5	4	simp1bi 1076	. . . . . . 7 |- ( rh -> i e. om )
6	5	adantl 482	. . . . . 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 30813	. . . . . 6 |- ( ( et /\ rh ) -> suc i e. m )
9	6, 8	jca 554	. . . . 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 30845	. . . . . . 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 2053	. . . . . . 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 207	. . . . . 6 |- ( ps' -> ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
14	13	imp32 449	. . . . 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 494	. . . 4 |- ( ( ta /\ ( et /\ rh ) ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
16	1, 15	simplbiim 659	. . 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 30840	. . . . 5 |- ( ( R FrSe A /\ ta /\ et ) -> Fun G )
19	18	bnj721 30827	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> Fun G )
20		bnj570.26	. . . . . 6 |- G = ( f u. { <. m , C >. } )
21	20	bnj931 30841	. . . . 5 |- f C_ G
22	21	a1i 11	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> f C_ G )
23		bnj667 30822	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( ta /\ et /\ rh ) )
24	2	bnj564 30814	. . . . . . 7 |- ( ta -> dom f = m )
25		eleq2 2690	. . . . . . . 8 |- ( dom f = m -> ( suc i e. dom f <-> suc i e. m ) )
26	25	biimpar 502	. . . . . . 7 |- ( ( dom f = m /\ suc i e. m ) -> suc i e. dom f )
27	24, 8, 26	syl2an 494	. . . . . 6 |- ( ( ta /\ ( et /\ rh ) ) -> suc i e. dom f )
28	27	3impb 1260	. . . . 5 |- ( ( ta /\ et /\ rh ) -> suc i e. dom f )
29	23, 28	syl 17	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> suc i e. dom f )
30	19, 22, 29	bnj1502 30918	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = ( f ` suc i ) )
31	2	simp1bi 1076	. . . . . . . . 9 |- ( ta -> f Fn m )
32		bnj252 30769	. . . . . . . . . . . . . 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 476	. . . . . . . . . . . . 13 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) -> m e. D )
34	7, 33	sylbi 207	. . . . . . . . . . . 12 |- ( et -> m e. D )
35		eldifi 3732	. . . . . . . . . . . . 13 |- ( m e. ( om \ { (/) } ) -> m e. om )
36		bnj570.3	. . . . . . . . . . . . 13 |- D = ( om \ { (/) } )
37	35, 36	eleq2s 2719	. . . . . . . . . . . 12 |- ( m e. D -> m e. om )
38		nnord 7073	. . . . . . . . . . . 12 |- ( m e. om -> Ord m )
39	34, 37, 38	3syl 18	. . . . . . . . . . 11 |- ( et -> Ord m )
40	39	adantr 481	. . . . . . . . . 10 |- ( ( et /\ rh ) -> Ord m )
41	40, 8	jca 554	. . . . . . . . 9 |- ( ( et /\ rh ) -> ( Ord m /\ suc i e. m ) )
42	31, 41	anim12i 590	. . . . . . . 8 |- ( ( ta /\ ( et /\ rh ) ) -> ( f Fn m /\ ( Ord m /\ suc i e. m ) ) )
43		fndm 5990	. . . . . . . . 9 |- ( f Fn m -> dom f = m )
44		elelsuc 5797	. . . . . . . . . 10 |- ( suc i e. m -> suc i e. suc m )
45		ordsucelsuc 7022	. . . . . . . . . . 11 |- ( Ord m -> ( i e. m <-> suc i e. suc m ) )
46	45	biimpar 502	. . . . . . . . . 10 |- ( ( Ord m /\ suc i e. suc m ) -> i e. m )
47	44, 46	sylan2 491	. . . . . . . . 9 |- ( ( Ord m /\ suc i e. m ) -> i e. m )
48	43, 47	anim12i 590	. . . . . . . 8 |- ( ( f Fn m /\ ( Ord m /\ suc i e. m ) ) -> ( dom f = m /\ i e. m ) )
49		eleq2 2690	. . . . . . . . 9 |- ( dom f = m -> ( i e. dom f <-> i e. m ) )
50	49	biimpar 502	. . . . . . . 8 |- ( ( dom f = m /\ i e. m ) -> i e. dom f )
51	42, 48, 50	3syl 18	. . . . . . 7 |- ( ( ta /\ ( et /\ rh ) ) -> i e. dom f )
52	51	3impb 1260	. . . . . 6 |- ( ( ta /\ et /\ rh ) -> i e. dom f )
53	23, 52	syl 17	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> i e. dom f )
54	19, 22, 53	bnj1502 30918	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` i ) = ( f ` i ) )
55	54	iuneq1d 4545	. . 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 2666	. 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 2674	1 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = K )

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   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   dom cdm 5114   Ord word 5722   suc csuc 5725   Fun wfun 5882   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

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:  bnj571  30976