Theorem bnj1145 33777

Description: Technical lemma for bnj69 33794. 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 33712	. 2 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
6		ss2iun 4331	. . . 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 33762	. . . . . 6 |- ( f e. B <-> E. n ch )
9	2	bnj1095 33568	. . . . . . . . 9 |- ( ps -> A. i ps )
10	9, 7	bnj1096 33569	. . . . . . . 8 |- ( ch -> A. i ch )
11	3	bnj1098 33570	. . . . . . . . . . . . . . . . 17 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )
12	7	bnj1232 33590	. . . . . . . . . . . . . . . . . 18 |- ( ch -> n e. D )
13	12	3anim3i 1185	. . . . . . . . . . . . . . . . 17 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( i =/= (/) /\ i e. n /\ n e. D ) )
14	11, 13	bnj1101 33571	. . . . . . . . . . . . . . . 16 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) )
15		ancl 546	. . . . . . . . . . . . . . . 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 33504	. . . . . . . . . . . . . . 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 312	. . . . . . . . . . . . . . . 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 1654	. . . . . . . . . . . . . . 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 209	. . . . . . . . . . . . . 14 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> th )
21		bnj213 33668	. . . . . . . . . . . . . . . 16 |- pred ( y , A , R ) C_ A
22	21	bnj226 33517	. . . . . . . . . . . . . . 15 |- U_ y e. ( f ` j ) pred ( y , A , R ) C_ A
23		simpr 461	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. n /\ i = suc j ) -> i = suc j )
24	17, 23	bnj833 33544	. . . . . . . . . . . . . . . . . 18 |- ( th -> i = suc j )
25		simp2 998	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. n )
26	12	3ad2ant3 1020	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> n e. D )
27	3	bnj923 33554	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. D -> n e. om )
28		elnn 6695	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( i e. n /\ n e. om ) -> i e. om )
29	27, 28	sylan2 474	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. n /\ n e. D ) -> i e. om )
30	25, 26, 29	syl2anc 661	. . . . . . . . . . . . . . . . . . 19 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. om )
31	17, 30	bnj832 33543	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. om )
32		vex 3098	. . . . . . . . . . . . . . . . . . . 20 |- j e. _V
33	32	bnj216 33515	. . . . . . . . . . . . . . . . . . 19 |- ( i = suc j -> j e. i )
34		elnn 6695	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. i /\ i e. om ) -> j e. om )
35	33, 34	sylan 471	. . . . . . . . . . . . . . . . . 18 |- ( ( i = suc j /\ i e. om ) -> j e. om )
36	24, 31, 35	syl2anc 661	. . . . . . . . . . . . . . . . 17 |- ( th -> j e. om )
37	17, 25	bnj832 33543	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. n )
38	24, 37	eqeltrrd 2532	. . . . . . . . . . . . . . . . 17 |- ( th -> suc j e. n )
39	2	bnj589 33695	. . . . . . . . . . . . . . . . . . . . . . 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 194	. . . . . . . . . . . . . . . . . . . . . 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 33541	. . . . . . . . . . . . . . . . . . . . 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 2809	. . . . . . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . . . . . 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 195	. . . . . . . . . . . . . . . . . . 19 |- ( ch -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
45	44	3ad2ant3 1020	. . . . . . . . . . . . . . . . . 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 33543	. . . . . . . . . . . . . . . . 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 45	. . . . . . . . . . . . . . . 16 |- ( th -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
48		fveq2 5856	. . . . . . . . . . . . . . . . . 18 |- ( i = suc j -> ( f ` i ) = ( f ` suc j ) )
49	48	eqeq1d 2445	. . . . . . . . . . . . . . . . 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 16	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . 15 |- ( th -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
52	22, 51	bnj1262 33597	. . . . . . . . . . . . . 14 |- ( th -> ( f ` i ) C_ A )
53	20, 52	bnj1023 33567	. . . . . . . . . . . . 13 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A )
54		3anass 978	. . . . . . . . . . . . . . 15 |- ( ( i =/= (/) /\ i e. n /\ ch ) <-> ( i =/= (/) /\ ( i e. n /\ ch ) ) )
55	54	imbi1i 325	. . . . . . . . . . . . . 14 |- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) )
56	55	exbii 1654	. . . . . . . . . . . . 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 208	. . . . . . . . . . . 12 |- E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
58	1	biimpi 194	. . . . . . . . . . . . . . 15 |- ( ph -> ( f ` (/) ) = pred ( X , A , R ) )
59	7, 58	bnj771 33550	. . . . . . . . . . . . . 14 |- ( ch -> ( f ` (/) ) = pred ( X , A , R ) )
60		fveq2 5856	. . . . . . . . . . . . . . 15 |- ( i = (/) -> ( f ` i ) = ( f ` (/) ) )
61		bnj213 33668	. . . . . . . . . . . . . . . 16 |- pred ( X , A , R ) C_ A
62		sseq1 3510	. . . . . . . . . . . . . . . 16 |- ( ( f ` (/) ) = pred ( X , A , R ) -> ( ( f ` (/) ) C_ A <-> pred ( X , A , R ) C_ A ) )
63	61, 62	mpbiri 233	. . . . . . . . . . . . . . 15 |- ( ( f ` (/) ) = pred ( X , A , R ) -> ( f ` (/) ) C_ A )
64		sseq1 3510	. . . . . . . . . . . . . . . 16 |- ( ( f ` i ) = ( f ` (/) ) -> ( ( f ` i ) C_ A <-> ( f ` (/) ) C_ A ) )
65	64	biimpar 485	. . . . . . . . . . . . . . 15 |- ( ( ( f ` i ) = ( f ` (/) ) /\ ( f ` (/) ) C_ A ) -> ( f ` i ) C_ A )
66	60, 63, 65	syl2an 477	. . . . . . . . . . . . . 14 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( f ` i ) C_ A )
67	59, 66	sylan2 474	. . . . . . . . . . . . 13 |- ( ( i = (/) /\ ch ) -> ( f ` i ) C_ A )
68	67	adantrl 715	. . . . . . . . . . . 12 |- ( ( i = (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
69	57, 68	bnj1109 33573	. . . . . . . . . . 11 |- E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
70		19.9v 1741	. . . . . . . . . . 11 |- ( E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) )
71	69, 70	mpbi 208	. . . . . . . . . 10 |- ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
72	71	expcom 435	. . . . . . . . 9 |- ( ch -> ( i e. n -> ( f ` i ) C_ A ) )
73		fndm 5670	. . . . . . . . . . 11 |- ( f Fn n -> dom f = n )
74	7, 73	bnj770 33549	. . . . . . . . . 10 |- ( ch -> dom f = n )
75		eleq2 2516	. . . . . . . . . . 11 |- ( dom f = n -> ( i e. dom f <-> i e. n ) )
76	75	imbi1d 317	. . . . . . . . . 10 |- ( dom f = n -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) )
77	74, 76	syl 16	. . . . . . . . 9 |- ( ch -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) )
78	72, 77	mpbird 232	. . . . . . . 8 |- ( ch -> ( i e. dom f -> ( f ` i ) C_ A ) )
79	10, 78	hbralrimi 2839	. . . . . . 7 |- ( ch -> A. i e. dom f ( f ` i ) C_ A )
80	79	exlimiv 1709	. . . . . 6 |- ( E. n ch -> A. i e. dom f ( f ` i ) C_ A )
81	8, 80	sylbi 195	. . . . 5 |- ( f e. B -> A. i e. dom f ( f ` i ) C_ A )
82		ss2iun 4331	. . . . . 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 33577	. . . . . 6 |- U_ i e. dom f A C_ A
84	82, 83	syl6ss 3501	. . . . 5 |- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ A )
85	81, 84	syl 16	. . . 4 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ A )
86	6, 85	mprg 2806	. . 3 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A
87	4	bnj1317 33608	. . . 4 |- ( w e. B -> A. f w e. B )
88	87	bnj1146 33578	. . 3 |- U_ f e. B A C_ A
89	86, 88	sstri 3498	. 2 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ A
90	5, 89	eqsstri 3519	1 |- trCl ( X , A , R ) C_ A

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 974   = wceq 1383   E.wex 1599   e. wcel 1804   {cab 2428   =/= wne 2638   A.wral 2793   E.wrex 2794   \ cdif 3458   C_ wss 3461   (/)c0 3770   {csn 4014   U_ciun 4315   suc csuc 4870   dom cdm 4989   Fn wfn 5573   ` cfv 5578   omcom 6685   /\ w-bnj17 33466   predc-bnj14 33468   trClc-bnj18 33474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-tr 4531  df-eprel 4781  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-iota 5541  df-fn 5581  df-fv 5586  df-om 6686  df-bnj17 33467  df-bnj14 33469  df-bnj18 33475

This theorem is referenced by:  bnj1147  33778