Theorem bnj570 29545

Description: Technical lemma for bnj852 29561. 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 29336	. . . 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 1022	. . . . 5 |- ( ta -> ps' )
4		bnj570.21	. . . . . . . 8 |- ( rh <-> ( i e. om /\ suc i e. n /\ m =/= suc i ) )
5	4	simp1bi 1020	. . . . . . 7 |- ( rh -> i e. om )
6	5	adantl 467	. . . . . 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 29382	. . . . . 6 |- ( ( et /\ rh ) -> suc i e. m )
9	6, 8	jca 534	. . . . 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 29415	. . . . . . 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 1909	. . . . . . 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 198	. . . . . 6 |- ( ps' -> ( i e. om -> ( suc i e. m -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
14	13	imp32 434	. . . . 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 479	. . . 4 |- ( ( ta /\ ( et /\ rh ) ) -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) )
16	1, 15	bnj833 29398	. . 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 29410	. . . . 5 |- ( ( R FrSe A /\ ta /\ et ) -> Fun G )
19	18	bnj721 29396	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> Fun G )
20		bnj570.26	. . . . . 6 |- G = ( f u. { <. m , C >. } )
21	20	bnj931 29411	. . . . 5 |- f C_ G
22	21	a1i 11	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> f C_ G )
23		bnj667 29391	. . . . 5 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( ta /\ et /\ rh ) )
24	2	bnj564 29383	. . . . . . 7 |- ( ta -> dom f = m )
25		eleq2 2493	. . . . . . . 8 |- ( dom f = m -> ( suc i e. dom f <-> suc i e. m ) )
26	25	biimpar 487	. . . . . . 7 |- ( ( dom f = m /\ suc i e. m ) -> suc i e. dom f )
27	24, 8, 26	syl2an 479	. . . . . 6 |- ( ( ta /\ ( et /\ rh ) ) -> suc i e. dom f )
28	27	3impb 1201	. . . . 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 29488	. . 3 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = ( f ` suc i ) )
31	2	simp1bi 1020	. . . . . . . . 9 |- ( ta -> f Fn m )
32		bnj252 29337	. . . . . . . . . . . . . 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 461	. . . . . . . . . . . . 13 |- ( ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) -> m e. D )
34	7, 33	sylbi 198	. . . . . . . . . . . 12 |- ( et -> m e. D )
35		eldifi 3584	. . . . . . . . . . . . 13 |- ( m e. ( om \ { (/) } ) -> m e. om )
36		bnj570.3	. . . . . . . . . . . . 13 |- D = ( om \ { (/) } )
37	35, 36	eleq2s 2528	. . . . . . . . . . . 12 |- ( m e. D -> m e. om )
38		nnord 6705	. . . . . . . . . . . 12 |- ( m e. om -> Ord m )
39	34, 37, 38	3syl 18	. . . . . . . . . . 11 |- ( et -> Ord m )
40	39	adantr 466	. . . . . . . . . 10 |- ( ( et /\ rh ) -> Ord m )
41	40, 8	jca 534	. . . . . . . . 9 |- ( ( et /\ rh ) -> ( Ord m /\ suc i e. m ) )
42	31, 41	anim12i 568	. . . . . . . 8 |- ( ( ta /\ ( et /\ rh ) ) -> ( f Fn m /\ ( Ord m /\ suc i e. m ) ) )
43		fndm 5684	. . . . . . . . 9 |- ( f Fn m -> dom f = m )
44		elelsuc 5505	. . . . . . . . . 10 |- ( suc i e. m -> suc i e. suc m )
45		ordsucelsuc 6654	. . . . . . . . . . 11 |- ( Ord m -> ( i e. m <-> suc i e. suc m ) )
46	45	biimpar 487	. . . . . . . . . 10 |- ( ( Ord m /\ suc i e. suc m ) -> i e. m )
47	44, 46	sylan2 476	. . . . . . . . 9 |- ( ( Ord m /\ suc i e. m ) -> i e. m )
48	43, 47	anim12i 568	. . . . . . . 8 |- ( ( f Fn m /\ ( Ord m /\ suc i e. m ) ) -> ( dom f = m /\ i e. m ) )
49		eleq2 2493	. . . . . . . . 9 |- ( dom f = m -> ( i e. dom f <-> i e. m ) )
50	49	biimpar 487	. . . . . . . 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 1201	. . . . . 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 29488	. . . 4 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` i ) = ( f ` i ) )
55	54	iuneq1d 4318	. . 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 2471	. 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 2479	1 |- ( ( R FrSe A /\ ta /\ et /\ rh ) -> ( G ` suc i ) = K )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   A.wal 1435   = wceq 1437   e. wcel 1867   =/= wne 2616   A.wral 2773   \ cdif 3430   u. cun 3431   C_ wss 3433   (/)c0 3758   {csn 3993   <.cop 3999   U_ciun 4293   dom cdm 4845   Ord word 5432   suc csuc 5435   Fun wfun 5586   Fn wfn 5587   ` cfv 5592   omcom 6697   /\ w-bnj17 29320   predc-bnj14 29322   FrSe w-bnj15 29326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-fv 5600  df-om 6698  df-bnj17 29321

This theorem is referenced by:  bnj571  29546