Theorem bnj1408 29066

Description: Technical lemma for bnj1414 29067. 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
bnj1408.1	|- B = ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
bnj1408.2	|- C = ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
bnj1408.3	|- ( th <-> ( R FrSe A /\ X e. A ) )
bnj1408.4	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )

Assertion

Ref	Expression
bnj1408	|- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = B )

Distinct variable groups:   y, A   y, R   y, X

Allowed substitution hints:   th( y)   ta( y)   B( y)   C( y)

Proof of Theorem bnj1408

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1408.3	. . . 4 |- ( th <-> ( R FrSe A /\ X e. A ) )
2	1	biimpri 197	. . 3 |- ( ( R FrSe A /\ X e. A ) -> th )
3		bnj1408.1	. . . . 5 |- B = ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
4	3	bnj1413 29065	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> B e. _V )
5		simplll 734	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> R FrSe A )
6		bnj213 28914	. . . . . . . . . . 11 |- pred ( X , A , R ) C_ A
7	6	sseli 3176	. . . . . . . . . 10 |- ( z e. pred ( X , A , R ) -> z e. A )
8	7	adantl 452	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> z e. A )
9		bnj906 28962	. . . . . . . . 9 |- ( ( R FrSe A /\ z e. A ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
10	5, 8, 9	syl2anc 642	. . . . . . . 8 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
11		bnj1318 29055	. . . . . . . . . . 11 |- ( y = z -> trCl ( y , A , R ) = trCl ( z , A , R ) )
12	11	ssiun2s 3946	. . . . . . . . . 10 |- ( z e. pred ( X , A , R ) -> trCl ( z , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
13		ssun4 3341	. . . . . . . . . . 11 |- ( trCl ( z , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) -> trCl ( z , A , R ) C_ ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) )
14	13, 3	syl6sseqr 3225	. . . . . . . . . 10 |- ( trCl ( z , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) -> trCl ( z , A , R ) C_ B )
15	12, 14	syl 15	. . . . . . . . 9 |- ( z e. pred ( X , A , R ) -> trCl ( z , A , R ) C_ B )
16	15	adantl 452	. . . . . . . 8 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> trCl ( z , A , R ) C_ B )
17	10, 16	sstrd 3189	. . . . . . 7 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> pred ( z , A , R ) C_ B )
18		simpr 447	. . . . . . . . . . 11 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
19	18	bnj1405 28869	. . . . . . . . . 10 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> E. y e. pred ( X , A , R ) z e. trCl ( y , A , R ) )
20		biid 227	. . . . . . . . . 10 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) <-> ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) )
21		nfv 1605	. . . . . . . . . . . . 13 |- F/ y ( R FrSe A /\ X e. A )
22		nfcv 2419	. . . . . . . . . . . . . . . 16 |- F/_ y pred ( X , A , R )
23		nfiu1 3933	. . . . . . . . . . . . . . . 16 |- F/_ y U_ y e. pred ( X , A , R ) trCl ( y , A , R )
24	22, 23	nfun 3331	. . . . . . . . . . . . . . 15 |- F/_ y ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
25	3, 24	nfcxfr 2416	. . . . . . . . . . . . . 14 |- F/_ y B
26	25	nfcri 2413	. . . . . . . . . . . . 13 |- F/ y z e. B
27	21, 26	nfan 1771	. . . . . . . . . . . 12 |- F/ y ( ( R FrSe A /\ X e. A ) /\ z e. B )
28	23	nfcri 2413	. . . . . . . . . . . 12 |- F/ y z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R )
29	27, 28	nfan 1771	. . . . . . . . . . 11 |- F/ y ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
30	29	nfri 1742	. . . . . . . . . 10 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> A. y ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) )
31	19, 20, 30	bnj1521 28883	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> E. y ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) )
32		simplll 734	. . . . . . . . . . . . 13 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> R FrSe A )
33	32	3ad2ant1 976	. . . . . . . . . . . 12 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> R FrSe A )
34		bnj1147 29024	. . . . . . . . . . . . 13 |- trCl ( y , A , R ) C_ A
35		simp3 957	. . . . . . . . . . . . 13 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> z e. trCl ( y , A , R ) )
36	34, 35	bnj1213 28831	. . . . . . . . . . . 12 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> z e. A )
37	33, 36, 9	syl2anc 642	. . . . . . . . . . 11 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
38		simp2 956	. . . . . . . . . . . . 13 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> y e. pred ( X , A , R ) )
39	6, 38	bnj1213 28831	. . . . . . . . . . . 12 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> y e. A )
40		bnj1125 29022	. . . . . . . . . . . 12 |- ( ( R FrSe A /\ y e. A /\ z e. trCl ( y , A , R ) ) -> trCl ( z , A , R ) C_ trCl ( y , A , R ) )
41	33, 39, 35, 40	syl3anc 1182	. . . . . . . . . . 11 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> trCl ( z , A , R ) C_ trCl ( y , A , R ) )
42	37, 41	sstrd 3189	. . . . . . . . . 10 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> pred ( z , A , R ) C_ trCl ( y , A , R ) )
43		ssiun2 3945	. . . . . . . . . . . 12 |- ( y e. pred ( X , A , R ) -> trCl ( y , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
44	43	3ad2ant2 977	. . . . . . . . . . 11 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> trCl ( y , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
45		ssun4 3341	. . . . . . . . . . . 12 |- ( trCl ( y , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) -> trCl ( y , A , R ) C_ ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) )
46	45, 3	syl6sseqr 3225	. . . . . . . . . . 11 |- ( trCl ( y , A , R ) C_ U_ y e. pred ( X , A , R ) trCl ( y , A , R ) -> trCl ( y , A , R ) C_ B )
47	44, 46	syl 15	. . . . . . . . . 10 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> trCl ( y , A , R ) C_ B )
48	42, 47	sstrd 3189	. . . . . . . . 9 |- ( ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) /\ y e. pred ( X , A , R ) /\ z e. trCl ( y , A , R ) ) -> pred ( z , A , R ) C_ B )
49	31, 48	bnj593 28774	. . . . . . . 8 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> E. y pred ( z , A , R ) C_ B )
50		nfcv 2419	. . . . . . . . . 10 |- F/_ y pred ( z , A , R )
51	50, 25	nfss 3173	. . . . . . . . 9 |- F/ y pred ( z , A , R ) C_ B
52	51	nfri 1742	. . . . . . . 8 |- ( pred ( z , A , R ) C_ B -> A. y pred ( z , A , R ) C_ B )
53	49, 52	bnj1397 28867	. . . . . . 7 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) -> pred ( z , A , R ) C_ B )
54		simpr 447	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> z e. B )
55	3	bnj1138 28820	. . . . . . . 8 |- ( z e. B <-> ( z e. pred ( X , A , R ) \/ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) )
56	54, 55	sylib 188	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> ( z e. pred ( X , A , R ) \/ z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) )
57	17, 53, 56	mpjaodan 761	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> pred ( z , A , R ) C_ B )
58	57	ralrimiva 2626	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> A. z e. B pred ( z , A , R ) C_ B )
59		df-bnj19 28722	. . . . 5 |- ( TrFo ( B , A , R ) <-> A. z e. B pred ( z , A , R ) C_ B )
60	58, 59	sylibr 203	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> TrFo ( B , A , R ) )
61	3	bnj931 28802	. . . . 5 |- pred ( X , A , R ) C_ B
62	61	a1i 10	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ B )
63		bnj1408.4	. . . 4 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
64	4, 60, 62, 63	syl3anbrc 1136	. . 3 |- ( ( R FrSe A /\ X e. A ) -> ta )
65	1, 63	bnj1124 29018	. . 3 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )
66	2, 64, 65	syl2anc 642	. 2 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) C_ B )
67		bnj906 28962	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
68		iunss1 3916	. . . . 5 |- ( pred ( X , A , R ) C_ trCl ( X , A , R ) -> U_ y e. pred ( X , A , R ) trCl ( y , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
69		unss2 3346	. . . . 5 |- ( U_ y e. pred ( X , A , R ) trCl ( y , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) -> ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) C_ ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) )
70	67, 68, 69	3syl 18	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) ) C_ ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) )
71		bnj1408.2	. . . 4 |- C = ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
72	70, 3, 71	3sstr4g 3219	. . 3 |- ( ( R FrSe A /\ X e. A ) -> B C_ C )
73		biid 227	. . . 4 |- ( ( R FrSe A /\ X e. A ) <-> ( R FrSe A /\ X e. A ) )
74		biid 227	. . . 4 |- ( ( C e. _V /\ TrFo ( C , A , R ) /\ pred ( X , A , R ) C_ C ) <-> ( C e. _V /\ TrFo ( C , A , R ) /\ pred ( X , A , R ) C_ C ) )
75	71, 73, 74	bnj1136 29027	. . 3 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = C )
76	72, 75	sseqtr4d 3215	. 2 |- ( ( R FrSe A /\ X e. A ) -> B C_ trCl ( X , A , R ) )
77	66, 76	eqssd 3196	1 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = B )

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   /\ w3a 934   = wceq 1623   e. wcel 1684   A.wral 2543   _Vcvv 2788   u. cun 3150   C_ wss 3152   U_ciun 3905   predc-bnj14 28713   FrSe w-bnj15 28717   trClc-bnj18 28719   TrFow-bnj19 28721

This theorem is referenced by:  bnj1414  29067

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28712  df-bnj14 28714  df-bnj13 28716  df-bnj15 28718  df-bnj18 28720  df-bnj19 28722