Theorem bnj1145 31818

Description: Technical lemma for bnj69 31835. 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 31753	. 2 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
6		ss2iun 4183	. . . 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 31803	. . . . . 6 |- ( f e. B <-> E. n ch )
9	2	bnj1095 31609	. . . . . . . . . 10 |- ( ps -> A. i ps )
10	9, 7	bnj1096 31610	. . . . . . . . 9 |- ( ch -> A. i ch )
11	10	nfi 1601	. . . . . . . 8 |- F/ i ch
12	3	bnj1098 31611	. . . . . . . . . . . . . . . . 17 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )
13	7	bnj1232 31631	. . . . . . . . . . . . . . . . . 18 |- ( ch -> n e. D )
14	13	3anim3i 1170	. . . . . . . . . . . . . . . . 17 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( i =/= (/) /\ i e. n /\ n e. D ) )
15	12, 14	bnj1101 31612	. . . . . . . . . . . . . . . 16 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) )
16		ancl 543	. . . . . . . . . . . . . . . 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 ) ) ) )
17	15, 16	bnj101 31546	. . . . . . . . . . . . . . 15 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) )
18		bnj1145.6	. . . . . . . . . . . . . . . . 17 |- ( th <-> ( ( i =/= (/) /\ i e. n /\ ch ) /\ ( j e. n /\ i = suc j ) ) )
19	18	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 ) ) ) )
20	19	exbii 1639	. . . . . . . . . . . . . . 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 ) ) ) )
21	17, 20	mpbir 209	. . . . . . . . . . . . . 14 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> th )
22		bnj213 31709	. . . . . . . . . . . . . . . 16 |- pred ( y , A , R ) C_ A
23	22	bnj226 31559	. . . . . . . . . . . . . . 15 |- U_ y e. ( f ` j ) pred ( y , A , R ) C_ A
24		simpr 458	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. n /\ i = suc j ) -> i = suc j )
25	18, 24	bnj833 31585	. . . . . . . . . . . . . . . . . 18 |- ( th -> i = suc j )
26		simp2 984	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. n )
27	13	3ad2ant3 1006	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> n e. D )
28	3	bnj923 31595	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. D -> n e. om )
29		elnn 6485	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( i e. n /\ n e. om ) -> i e. om )
30	28, 29	sylan2 471	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. n /\ n e. D ) -> i e. om )
31	26, 27, 30	syl2anc 656	. . . . . . . . . . . . . . . . . . 19 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. om )
32	18, 31	bnj832 31584	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. om )
33		vex 2973	. . . . . . . . . . . . . . . . . . . 20 |- j e. _V
34	33	bnj216 31557	. . . . . . . . . . . . . . . . . . 19 |- ( i = suc j -> j e. i )
35		elnn 6485	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. i /\ i e. om ) -> j e. om )
36	34, 35	sylan 468	. . . . . . . . . . . . . . . . . 18 |- ( ( i = suc j /\ i e. om ) -> j e. om )
37	25, 32, 36	syl2anc 656	. . . . . . . . . . . . . . . . 17 |- ( th -> j e. om )
38	18, 26	bnj832 31584	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. n )
39	25, 38	eqeltrrd 2516	. . . . . . . . . . . . . . . . 17 |- ( th -> suc j e. n )
40	2	bnj589 31736	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ps <-> A. j e. om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
41	40	biimpi 194	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ps -> A. j e. om ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) )
42	41	bnj708 31582	. . . . . . . . . . . . . . . . . . . . 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 ) ) )
43		rsp 2774	. . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
44	42, 43	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 ) ) ) )
45	7, 44	sylbi 195	. . . . . . . . . . . . . . . . . . 19 |- ( ch -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
46	45	3ad2ant3 1006	. . . . . . . . . . . . . . . . . 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 ) ) ) )
47	18, 46	bnj832 31584	. . . . . . . . . . . . . . . . 17 |- ( th -> ( j e. om -> ( suc j e. n -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) ) ) )
48	37, 39, 47	mp2d 45	. . . . . . . . . . . . . . . 16 |- ( th -> ( f ` suc j ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
49		fveq2 5688	. . . . . . . . . . . . . . . . . 18 |- ( i = suc j -> ( f ` i ) = ( f ` suc j ) )
50	49	eqeq1d 2449	. . . . . . . . . . . . . . . . 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 ) ) )
51	25, 50	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 ) ) )
52	48, 51	mpbird 232	. . . . . . . . . . . . . . 15 |- ( th -> ( f ` i ) = U_ y e. ( f ` j ) pred ( y , A , R ) )
53	23, 52	bnj1262 31638	. . . . . . . . . . . . . 14 |- ( th -> ( f ` i ) C_ A )
54	21, 53	bnj1023 31608	. . . . . . . . . . . . 13 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A )
55		3anass 964	. . . . . . . . . . . . . . 15 |- ( ( i =/= (/) /\ i e. n /\ ch ) <-> ( i =/= (/) /\ ( i e. n /\ ch ) ) )
56	55	imbi1i 325	. . . . . . . . . . . . . 14 |- ( ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) )
57	56	exbii 1639	. . . . . . . . . . . . 13 |- ( E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A ) )
58	54, 57	mpbi 208	. . . . . . . . . . . 12 |- E. j ( ( i =/= (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
59	1	biimpi 194	. . . . . . . . . . . . . . 15 |- ( ph -> ( f ` (/) ) = pred ( X , A , R ) )
60	7, 59	bnj771 31591	. . . . . . . . . . . . . 14 |- ( ch -> ( f ` (/) ) = pred ( X , A , R ) )
61		fveq2 5688	. . . . . . . . . . . . . . 15 |- ( i = (/) -> ( f ` i ) = ( f ` (/) ) )
62		bnj213 31709	. . . . . . . . . . . . . . . 16 |- pred ( X , A , R ) C_ A
63		sseq1 3374	. . . . . . . . . . . . . . . 16 |- ( ( f ` (/) ) = pred ( X , A , R ) -> ( ( f ` (/) ) C_ A <-> pred ( X , A , R ) C_ A ) )
64	62, 63	mpbiri 233	. . . . . . . . . . . . . . 15 |- ( ( f ` (/) ) = pred ( X , A , R ) -> ( f ` (/) ) C_ A )
65		sseq1 3374	. . . . . . . . . . . . . . . 16 |- ( ( f ` i ) = ( f ` (/) ) -> ( ( f ` i ) C_ A <-> ( f ` (/) ) C_ A ) )
66	65	biimpar 482	. . . . . . . . . . . . . . 15 |- ( ( ( f ` i ) = ( f ` (/) ) /\ ( f ` (/) ) C_ A ) -> ( f ` i ) C_ A )
67	61, 64, 66	syl2an 474	. . . . . . . . . . . . . 14 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( f ` i ) C_ A )
68	60, 67	sylan2 471	. . . . . . . . . . . . 13 |- ( ( i = (/) /\ ch ) -> ( f ` i ) C_ A )
69	68	adantrl 710	. . . . . . . . . . . 12 |- ( ( i = (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
70	58, 69	bnj1109 31614	. . . . . . . . . . 11 |- E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
71		19.9v 1722	. . . . . . . . . . 11 |- ( E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) <-> ( ( i e. n /\ ch ) -> ( f ` i ) C_ A ) )
72	70, 71	mpbi 208	. . . . . . . . . 10 |- ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
73	72	expcom 435	. . . . . . . . 9 |- ( ch -> ( i e. n -> ( f ` i ) C_ A ) )
74		fndm 5507	. . . . . . . . . . 11 |- ( f Fn n -> dom f = n )
75	7, 74	bnj770 31590	. . . . . . . . . 10 |- ( ch -> dom f = n )
76		eleq2 2502	. . . . . . . . . . 11 |- ( dom f = n -> ( i e. dom f <-> i e. n ) )
77	76	imbi1d 317	. . . . . . . . . 10 |- ( dom f = n -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) )
78	75, 77	syl 16	. . . . . . . . 9 |- ( ch -> ( ( i e. dom f -> ( f ` i ) C_ A ) <-> ( i e. n -> ( f ` i ) C_ A ) ) )
79	73, 78	mpbird 232	. . . . . . . 8 |- ( ch -> ( i e. dom f -> ( f ` i ) C_ A ) )
80	11, 79	ralrimi 2795	. . . . . . 7 |- ( ch -> A. i e. dom f ( f ` i ) C_ A )
81	80	exlimiv 1693	. . . . . 6 |- ( E. n ch -> A. i e. dom f ( f ` i ) C_ A )
82	8, 81	sylbi 195	. . . . 5 |- ( f e. B -> A. i e. dom f ( f ` i ) C_ A )
83		ss2iun 4183	. . . . . 6 |- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ U_ i e. dom f A )
84		bnj1143 31618	. . . . . 6 |- U_ i e. dom f A C_ A
85	83, 84	syl6ss 3365	. . . . 5 |- ( A. i e. dom f ( f ` i ) C_ A -> U_ i e. dom f ( f ` i ) C_ A )
86	82, 85	syl 16	. . . 4 |- ( f e. B -> U_ i e. dom f ( f ` i ) C_ A )
87	6, 86	mprg 2783	. . 3 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A
88	4	bnj1317 31649	. . . 4 |- ( w e. B -> A. f w e. B )
89	88	bnj1146 31619	. . 3 |- U_ f e. B A C_ A
90	87, 89	sstri 3362	. 2 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ A
91	5, 90	eqsstri 3383	1 |- trCl ( X , A , R ) C_ A

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 960   = wceq 1364   E.wex 1591   e. wcel 1761   {cab 2427   =/= wne 2604   A.wral 2713   E.wrex 2714   \ cdif 3322   C_ wss 3325   (/)c0 3634   {csn 3874   U_ciun 4168   suc csuc 4717   dom cdm 4836   Fn wfn 5410   ` cfv 5415   omcom 6475   /\ w-bnj17 31508   predc-bnj14 31510   trClc-bnj18 31516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-iota 5378  df-fn 5418  df-fv 5423  df-om 6476  df-bnj17 31509  df-bnj14 31511  df-bnj18 31517

This theorem is referenced by:  bnj1147  31819