Theorem bnj570 29709

Description: Technical lemma for bnj852 29725. 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 29500	. . . 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 1024	. . . . 5 |- ( ta -> ps' )
4		bnj570.21	. . . . . . . 8 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
5	4	simp1bi 1022	. . . . . . 7 |- ( rh -> i e. om )
6	5	adantl 468	. . . . . 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 29546	. . . . . 6 |- ( ( et /\ rh ) -> suc i e. m )
9	6, 8	jca 535	. . . . 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 29579	. . . . . . 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 1936	. . . . . . 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 199	. . . . . 6 |- ( ps' -> ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
14	13	imp32 435	. . . . 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 480	. . . 4 |- ( ( ta /\ ( et /\ rh ) ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
16	1, 15	bnj833 29562	. . 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 29574	. . . . 5 |- ( ( R FrSe A /\ ta /\ et ) -> Fun G )
19	18	bnj721 29560	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> Fun G )
20		bnj570.26	. . . . . 6 |- G = ( f u. { <. m , C >. } )
21	20	bnj931 29575	. . . . 5 |- f C_ G
22	21	a1i 11	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> f C_ G )
23		bnj667 29555	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( ta /\ et /\ rh ) )
24	2	bnj564 29547	. . . . . . 7 |- ( ta -> dom f = m )
25		eleq2 2517	. . . . . . . 8 |- ( dom f = m -> ( suc i e. dom f <-> suc i e. m ) )
26	25	biimpar 488	. . . . . . 7 |- ( ( dom f = m /\ suc i e. m ) -> suc i e. dom f )
27	24, 8, 26	syl2an 480	. . . . . 6 |- ( ( ta /\ ( et /\ rh ) ) -> suc i e. dom f )
28	27	3impb 1203	. . . . 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 29652	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = ( f ` suc i ) )
31	2	simp1bi 1022	. . . . . . . . 9 |- ( ta -> f Fn m )
32		bnj252 29501	. . . . . . . . . . . . . 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 462	. . . . . . . . . . . . 13 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) -> m e. D )
34	7, 33	sylbi 199	. . . . . . . . . . . 12 |- ( et -> m e. D )
35		eldifi 3554	. . . . . . . . . . . . 13 |- ( m e. ( om \ { (/) } ) -> m e. om )
36		bnj570.3	. . . . . . . . . . . . 13 |- D = ( om \ { (/) } )
37	35, 36	eleq2s 2546	. . . . . . . . . . . 12 |- ( m e. D -> m e. om )
38		nnord 6697	. . . . . . . . . . . 12 |- ( m e. om -> Ord m )
39	34, 37, 38	3syl 18	. . . . . . . . . . 11 |- ( et -> Ord m )
40	39	adantr 467	. . . . . . . . . 10 |- ( ( et /\ rh ) -> Ord m )
41	40, 8	jca 535	. . . . . . . . 9 |- ( ( et /\ rh ) -> ( Ord m /\ suc i e. m ) )
42	31, 41	anim12i 569	. . . . . . . 8 |- ( ( ta /\ ( et /\ rh ) ) -> ( f Fn m /\ ( Ord m /\ suc i e. m ) ) )
43		fndm 5673	. . . . . . . . 9 |- ( f Fn m -> dom f = m )
44		elelsuc 5494	. . . . . . . . . 10 |- ( suc i e. m -> suc i e. suc m )
45		ordsucelsuc 6646	. . . . . . . . . . 11 |- ( Ord m -> ( i e. m <-> suc i e. suc m ) )
46	45	biimpar 488	. . . . . . . . . 10 |- ( ( Ord m /\ suc i e. suc m ) -> i e. m )
47	44, 46	sylan2 477	. . . . . . . . 9 |- ( ( Ord m /\ suc i e. m ) -> i e. m )
48	43, 47	anim12i 569	. . . . . . . 8 |- ( ( f Fn m /\ ( Ord m /\ suc i e. m ) ) -> ( dom f = m /\ i e. m ) )
49		eleq2 2517	. . . . . . . . 9 |- ( dom f = m -> ( i e. dom f <-> i e. m ) )
50	49	biimpar 488	. . . . . . . 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 1203	. . . . . 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 29652	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` i ) = ( f ` i ) )
55	54	iuneq1d 4302	. . 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 2494	. 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 2502	1 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = K )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 984   A.wal 1441   = wceq 1443   e. wcel 1886   =/= wne 2621   A.wral 2736   \ cdif 3400   u. cun 3401   C_ wss 3403   (/)c0 3730   {csn 3967   <.cop 3973   U_ciun 4277   dom cdm 4833   Ord word 5421   suc csuc 5424   Fun wfun 5575   Fn wfn 5576   ` cfv 5581   omcom 6689   /\ w-bnj17 29484   predc-bnj14 29486   FrSe w-bnj15 29490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-res 4845  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-fv 5589  df-om 6690  df-bnj17 29485

This theorem is referenced by:  bnj571  29710