Theorem bnj1145 33137

Description: Technical lemma for bnj69 33154. 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 33072	. 2 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )
6		ss2iun 4341	. . . 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 33122	. . . . . 6 |- ( f e. B <-> E. n ch )
9	2	bnj1095 32928	. . . . . . . . . 10 |- ( ps -> A. i ps )
10	9, 7	bnj1096 32929	. . . . . . . . 9 |- ( ch -> A. i ch )
11	10	nfi 1606	. . . . . . . 8 |- F/ i ch
12	3	bnj1098 32930	. . . . . . . . . . . . . . . . 17 |- E. j ( ( i =/= (/) /\ i e. n /\ n e. D ) -> ( j e. n /\ i = suc j ) )
13	7	bnj1232 32950	. . . . . . . . . . . . . . . . . 18 |- ( ch -> n e. D )
14	13	3anim3i 1184	. . . . . . . . . . . . . . . . 17 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> ( i =/= (/) /\ i e. n /\ n e. D ) )
15	12, 14	bnj1101 32931	. . . . . . . . . . . . . . . 16 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( j e. n /\ i = suc j ) )
16		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 ) ) ) )
17	15, 16	bnj101 32865	. . . . . . . . . . . . . . 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 1644	. . . . . . . . . . . . . . 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 33028	. . . . . . . . . . . . . . . 16 |- pred ( y , A , R ) C_ A
23	22	bnj226 32878	. . . . . . . . . . . . . . 15 |- U_ y e. ( f ` j ) pred ( y , A , R ) C_ A
24		simpr 461	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. n /\ i = suc j ) -> i = suc j )
25	18, 24	bnj833 32904	. . . . . . . . . . . . . . . . . 18 |- ( th -> i = suc j )
26		simp2 997	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. n )
27	13	3ad2ant3 1019	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> n e. D )
28	3	bnj923 32914	. . . . . . . . . . . . . . . . . . . . 21 |- ( n e. D -> n e. om )
29		elnn 6689	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( i e. n /\ n e. om ) -> i e. om )
30	28, 29	sylan2 474	. . . . . . . . . . . . . . . . . . . 20 |- ( ( i e. n /\ n e. D ) -> i e. om )
31	26, 27, 30	syl2anc 661	. . . . . . . . . . . . . . . . . . 19 |- ( ( i =/= (/) /\ i e. n /\ ch ) -> i e. om )
32	18, 31	bnj832 32903	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. om )
33		vex 3116	. . . . . . . . . . . . . . . . . . . 20 |- j e. _V
34	33	bnj216 32876	. . . . . . . . . . . . . . . . . . 19 |- ( i = suc j -> j e. i )
35		elnn 6689	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. i /\ i e. om ) -> j e. om )
36	34, 35	sylan 471	. . . . . . . . . . . . . . . . . 18 |- ( ( i = suc j /\ i e. om ) -> j e. om )
37	25, 32, 36	syl2anc 661	. . . . . . . . . . . . . . . . 17 |- ( th -> j e. om )
38	18, 26	bnj832 32903	. . . . . . . . . . . . . . . . . 18 |- ( th -> i e. n )
39	25, 38	eqeltrrd 2556	. . . . . . . . . . . . . . . . 17 |- ( th -> suc j e. n )
40	2	bnj589 33055	. . . . . . . . . . . . . . . . . . . . . . 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 32901	. . . . . . . . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . . 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 1019	. . . . . . . . . . . . . . . . . 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 32903	. . . . . . . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . . . . 18 |- ( i = suc j -> ( f ` i ) = ( f ` suc j ) )
50	49	eqeq1d 2469	. . . . . . . . . . . . . . . . 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 32957	. . . . . . . . . . . . . 14 |- ( th -> ( f ` i ) C_ A )
54	21, 53	bnj1023 32927	. . . . . . . . . . . . 13 |- E. j ( ( i =/= (/) /\ i e. n /\ ch ) -> ( f ` i ) C_ A )
55		3anass 977	. . . . . . . . . . . . . . 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 1644	. . . . . . . . . . . . 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 32910	. . . . . . . . . . . . . 14 |- ( ch -> ( f ` (/) ) = pred ( X , A , R ) )
61		fveq2 5865	. . . . . . . . . . . . . . 15 |- ( i = (/) -> ( f ` i ) = ( f ` (/) ) )
62		bnj213 33028	. . . . . . . . . . . . . . . 16 |- pred ( X , A , R ) C_ A
63		sseq1 3525	. . . . . . . . . . . . . . . 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 3525	. . . . . . . . . . . . . . . 16 |- ( ( f ` i ) = ( f ` (/) ) -> ( ( f ` i ) C_ A <-> ( f ` (/) ) C_ A ) )
66	65	biimpar 485	. . . . . . . . . . . . . . 15 |- ( ( ( f ` i ) = ( f ` (/) ) /\ ( f ` (/) ) C_ A ) -> ( f ` i ) C_ A )
67	61, 64, 66	syl2an 477	. . . . . . . . . . . . . 14 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( f ` i ) C_ A )
68	60, 67	sylan2 474	. . . . . . . . . . . . 13 |- ( ( i = (/) /\ ch ) -> ( f ` i ) C_ A )
69	68	adantrl 715	. . . . . . . . . . . 12 |- ( ( i = (/) /\ ( i e. n /\ ch ) ) -> ( f ` i ) C_ A )
70	58, 69	bnj1109 32933	. . . . . . . . . . 11 |- E. j ( ( i e. n /\ ch ) -> ( f ` i ) C_ A )
71		19.9v 1728	. . . . . . . . . . 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 5679	. . . . . . . . . . 11 |- ( f Fn n -> dom f = n )
75	7, 74	bnj770 32909	. . . . . . . . . 10 |- ( ch -> dom f = n )
76		eleq2 2540	. . . . . . . . . . 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 2864	. . . . . . 7 |- ( ch -> A. i e. dom f ( f ` i ) C_ A )
81	80	exlimiv 1698	. . . . . 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 4341	. . . . . 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 32937	. . . . . 6 |- U_ i e. dom f A C_ A
85	83, 84	syl6ss 3516	. . . . 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 2827	. . 3 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ U_ f e. B A
88	4	bnj1317 32968	. . . 4 |- ( w e. B -> A. f w e. B )
89	88	bnj1146 32938	. . 3 |- U_ f e. B A C_ A
90	87, 89	sstri 3513	. 2 |- U_ f e. B U_ i e. dom f ( f ` i ) C_ A
91	5, 90	eqsstri 3534	1 |- trCl ( X , A , R ) C_ A

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   =/= wne 2662   A.wral 2814   E.wrex 2815   \ cdif 3473   C_ wss 3476   (/)c0 3785   {csn 4027   U_ciun 4325   suc csuc 4880   dom cdm 4999   Fn wfn 5582   ` cfv 5587   omcom 6679   /\ w-bnj17 32827   predc-bnj14 32829   trClc-bnj18 32835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-iota 5550  df-fn 5590  df-fv 5595  df-om 6680  df-bnj17 32828  df-bnj14 32830  df-bnj18 32836

This theorem is referenced by:  bnj1147  33138