Theorem bnj1145 31061

Description: Technical lemma for bnj69 31078. 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
bnj1145.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1145.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj1145.3	|- D = ( om \ { (/) } )
bnj1145.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
bnj1145.5	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1145.6	|- ( th <-> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) )

Assertion

Ref	Expression
bnj1145	|- trCl ( X , A , R ) C_ A

Distinct variable groups:   A, f, i, j, n, y   D, i, j   R, f, i, j, n, y   f, X, i, n, y   ch, j   ph, i

Allowed substitution hints:   ph( y, f, j, n)   ps( y, f, i, j, n)   ch( y, f, i, n)   th( y, f, i, j, n)   B( y, f, i, j, n)   D( y, f, n)   X( j)

Proof of Theorem bnj1145

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1145.1	. . 3 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
2		bnj1145.2	. . 3 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
3		bnj1145.3	. . 3 |- D = ( om \ { (/) } )
4		bnj1145.4	. . 3 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5	1, 2, 3, 4	bnj882 30996	. 2 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
6		ss2iun 4536	. . . 4 |- ( A. f e. B U_ i e. dom f ( f ` i ) C_ A -> U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A )
7		bnj1145.5	. . . . . . 7 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
8	7, 4	bnj1083 31046	. . . . . 6 |- ( f e. B <-> E. n ch )
9	2	bnj1095 30852	. . . . . . . . 9 |- ( ps -> A. i ps )
10	9, 7	bnj1096 30853	. . . . . . . 8 |- ( ch -> A. i ch )
11	3	bnj1098 30854	. . . . . . . . . . . . . . . . 17 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )
12	7	bnj1232 30874	. . . . . . . . . . . . . . . . . 18 |- ( ch -> n e. D )
13	12	3anim3i 1250	. . . . . . . . . . . . . . . . 17 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( i =/= (/) /\ i e. n /\ n e. D ) )
14	11, 13	bnj1101 30855	. . . . . . . . . . . . . . . 16 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) )
15		ancl 569	. . . . . . . . . . . . . . . 16 |- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) ) -> ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) )
16	14, 15	bnj101 30789	. . . . . . . . . . . . . . 15 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) )
17		bnj1145.6	. . . . . . . . . . . . . . . . 17 |- ( th <-> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) )
18	17	imbi2i 326	. . . . . . . . . . . . . . . 16 |- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> th ) <-> ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) )
19	18	exbii 1774	. . . . . . . . . . . . . . 15 |- ( E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> th ) <-> E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) ) )
20	16, 19	mpbir 221	. . . . . . . . . . . . . 14 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> th )
21		bnj213 30952	. . . . . . . . . . . . . . . 16 |- pred ( y , A , R ) C_ A
22	21	bnj226 30802	. . . . . . . . . . . . . . 15 |- U_ y e. ( f ` j ) pred ( y , A , R ) C_ A
23		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. n /\ i = suc j ) -> i = suc j )
24	17, 23	simplbiim 659	. . . . . . . . . . . . . . . . . 18 |- ( th -> i = suc j )
25		simp2 1062	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. n )
26	12	3ad2ant3 1084	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> n e. D )
27	3	bnj923 30838	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. D -> n e. om )
28		elnn 7075	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( i e. n /\ n e. om ) -> i e. om )
29	27, 28	sylan2 491	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. n /\ n e. D ) -> i e. om )
30	25, 26, 29	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. om )
31	17, 30	bnj832 30828	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. om )
32		vex 3203	. . . . . . . . . . . . . . . . . . . 20 |- j e. _V
33	32	bnj216 30800	. . . . . . . . . . . . . . . . . . 19 |- ( i = suc j -> j e. i )
34		elnn 7075	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. i /\ i e. om ) -> j e. om )
35	33, 34	sylan 488	. . . . . . . . . . . . . . . . . 18 |- ( ( i = suc j /\ i e. om ) -> j e. om )
36	24, 31, 35	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( th -> j e. om )
37	17, 25	bnj832 30828	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. n )
38	24, 37	eqeltrrd 2702	. . . . . . . . . . . . . . . . 17 |- ( th -> suc j e. n )
39	2	bnj589 30979	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ps <-> A. j e. om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
40	39	biimpi 206	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ps -> A. j e. om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
41	40	bnj708 30826	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> A. j e. om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
42		rsp 2929	. . . . . . . . . . . . . . . . . . . . 21 |- ( A. j e. om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
43	41, 42	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ( n e. D /\ f Fn n /\ ph /\ ps ) -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
44	7, 43	sylbi 207	. . . . . . . . . . . . . . . . . . 19 |- ( ch -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
45	44	3ad2ant3 1084	. . . . . . . . . . . . . . . . . 18 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
46	17, 45	bnj832 30828	. . . . . . . . . . . . . . . . 17 |- ( th -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
47	36, 38, 46	mp2d 49	. . . . . . . . . . . . . . . 16 |- ( th -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
48		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( i = suc j -> ( f ` i ) = ( f ` suc j ) )
49	48	eqeq1d 2624	. . . . . . . . . . . . . . . . 17 |- ( i = suc j -> ( ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) <-> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
50	24, 49	syl 17	. . . . . . . . . . . . . . . 16 |- ( th -> ( ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) <-> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
51	47, 50	mpbird 247	. . . . . . . . . . . . . . 15 |- ( th -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
52	22, 51	bnj1262 30881	. . . . . . . . . . . . . 14 |- ( th -> ( f ` i ) C_ A )
53	20, 52	bnj1023 30851	. . . . . . . . . . . . 13 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A )
54		3anass 1042	. . . . . . . . . . . . . . 15 |- ( ( i =/= (/) /\ i e. n /\ ch ) <-> ( i =/= (/) /\ ( i e. n /\ ch ) ) )
55	54	imbi1i 339	. . . . . . . . . . . . . 14 |- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) )
56	55	exbii 1774	. . . . . . . . . . . . 13 |- ( E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) )
57	53, 56	mpbi 220	. . . . . . . . . . . 12 |- E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
58	1	biimpi 206	. . . . . . . . . . . . . . 15 |- ( ph -> ( f ` (/) ) = pred ( X , A , R ) )
59	7, 58	bnj771 30834	. . . . . . . . . . . . . 14 |- ( ch -> ( f ` (/) ) = pred ( X , A , R ) )
60		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( i = (/) -> ( f ` i ) = ( f ` (/) ) )
61		bnj213 30952	. . . . . . . . . . . . . . . 16 |- pred ( X , A , R ) C_ A
62		sseq1 3626	. . . . . . . . . . . . . . . 16 |- ( ( f ` (/) ) = pred ( X , A , R ) -> ( ( f ` (/) ) C_ A <-> pred ( X , A , R ) C_ A ) )
63	61, 62	mpbiri 248	. . . . . . . . . . . . . . 15 |- ( ( f ` (/) ) = pred ( X , A , R ) -> ( f ` (/) ) C_ A )
64		sseq1 3626	. . . . . . . . . . . . . . . 16 |- ( ( f ` i ) = ( f ` (/) ) -> ( ( f ` i ) C_ A <-> ( f ` (/) ) C_ A ) )
65	64	biimpar 502	. . . . . . . . . . . . . . 15 |- ( ( ( f ` i ) = ( f ` (/) ) /\ ( f ` (/) ) C_ A ) -> ( f ` i ) C_ A )
66	60, 63, 65	syl2an 494	. . . . . . . . . . . . . 14 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( f ` i ) C_ A )
67	59, 66	sylan2 491	. . . . . . . . . . . . 13 |- ( ( i = (/) /\ ch ) -> ( f ` i ) C_ A )
68	67	adantrl 752	. . . . . . . . . . . 12 |- ( ( i = (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
69	57, 68	bnj1109 30857	. . . . . . . . . . 11 |- E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
70		19.9v 1896	. . . . . . . . . . 11 |- ( E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) )
71	69, 70	mpbi 220	. . . . . . . . . 10 |- ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
72	71	expcom 451	. . . . . . . . 9 |- ( ch -> ( i e. n -> ( f ` i ) C_ A ) )
73		fndm 5990	. . . . . . . . . . 11 |- ( f Fn n -> dom f = n )
74	7, 73	bnj770 30833	. . . . . . . . . 10 |- ( ch -> dom f = n )
75		eleq2 2690	. . . . . . . . . . 11 |- ( dom f = n -> ( i e. dom f <-> i e. n ) )
76	75	imbi1d 331	. . . . . . . . . 10 |- ( dom f = n -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) )
77	74, 76	syl 17	. . . . . . . . 9 |- ( ch -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) )
78	72, 77	mpbird 247	. . . . . . . 8 |- ( ch -> ( i e. dom f -> ( f ` i ) C_ A ) )
79	10, 78	hbralrimi 2954	. . . . . . 7 |- ( ch -> A. i e. dom f ( f ` i ) C_ A )
80	79	exlimiv 1858	. . . . . 6 |- ( E. n ch -> A. i e. dom f ( f ` i ) C_ A )
81	8, 80	sylbi 207	. . . . 5 |- ( f e. B -> A. i e. dom f ( f ` i ) C_ A )
82		ss2iun 4536	. . . . . 6 |- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ U_ i e. dom f A )
83		bnj1143 30861	. . . . . 6 |- U_ i e. dom f A C_ A
84	82, 83	syl6ss 3615	. . . . 5 |- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ A )
85	81, 84	syl 17	. . . 4 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ A )
86	6, 85	mprg 2926	. . 3 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A
87	4	bnj1317 30892	. . . 4 |- ( w e. B -> A. f w e. B )
88	87	bnj1146 30862	. . 3 |- U_ f e. B A C_ A
89	86, 88	sstri 3612	. 2 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ A
90	5, 89	eqsstri 3635	1 |- trCl ( X , A , R ) C_ A

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   predc-bnj14 30754   trClc-bnj18 30760

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-pw 4160  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-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fn 5891  df-fv 5896  df-om 7066  df-bnj17 30753  df-bnj14 30755  df-bnj18 30761

This theorem is referenced by:  bnj1147  31062