Theorem bnj1408 31104

Description: Technical lemma for bnj1414 31105. 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 218	. . 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 31103	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> B e. _V )
5		simplll 798	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> R FrSe A )
6		bnj213 30952	. . . . . . . . . . 11 |- pred ( X , A , R ) C_ A
7	6	sseli 3599	. . . . . . . . . 10 |- ( z e. pred ( X , A , R ) -> z e. A )
8	7	adantl 482	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> z e. A )
9		bnj906 31000	. . . . . . . . 9 |- ( ( R FrSe A /\ z e. A ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
10	5, 8, 9	syl2anc 693	. . . . . . . 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 31093	. . . . . . . . . . 11 |- ( y = z -> trCl ( y , A , R ) = trCl ( z , A , R ) )
12	11	ssiun2s 4564	. . . . . . . . . 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 3779	. . . . . . . . . . 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 3652	. . . . . . . . . 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 17	. . . . . . . . 9 |- ( z e. pred ( X , A , R ) -> trCl ( z , A , R ) C_ B )
16	15	adantl 482	. . . . . . . 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 3613	. . . . . . 7 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> pred ( z , A , R ) C_ B )
18		simpr 477	. . . . . . . . . . 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 30907	. . . . . . . . . 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 251	. . . . . . . . . 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 1843	. . . . . . . . . . . . 13 |- F/ y ( R FrSe A /\ X e. A )
22		nfcv 2764	. . . . . . . . . . . . . . . 16 |- F/_ y pred ( X , A , R )
23		nfiu1 4550	. . . . . . . . . . . . . . . 16 |- F/_ y U_ y e. pred ( X , A , R ) trCl ( y , A , R )
24	22, 23	nfun 3769	. . . . . . . . . . . . . . 15 |- F/_ y ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
25	3, 24	nfcxfr 2762	. . . . . . . . . . . . . 14 |- F/_ y B
26	25	nfcri 2758	. . . . . . . . . . . . 13 |- F/ y z e. B
27	21, 26	nfan 1828	. . . . . . . . . . . 12 |- F/ y ( ( R FrSe A /\ X e. A ) /\ z e. B )
28	23	nfcri 2758	. . . . . . . . . . . 12 |- F/ y z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R )
29	27, 28	nfan 1828	. . . . . . . . . . 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	nf5ri 2065	. . . . . . . . . 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 30921	. . . . . . . . 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 798	. . . . . . . . . . . . 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 1082	. . . . . . . . . . . 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 31062	. . . . . . . . . . . . 13 |- trCl ( y , A , R ) C_ A
35		simp3 1063	. . . . . . . . . . . . 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 30869	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 1062	. . . . . . . . . . . . 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 30869	. . . . . . . . . . . 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 31060	. . . . . . . . . . . 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 1326	. . . . . . . . . . 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 3613	. . . . . . . . . 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 4563	. . . . . . . . . . . 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 1083	. . . . . . . . . . 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 3779	. . . . . . . . . . . 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 3652	. . . . . . . . . . 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 17	. . . . . . . . . 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 3613	. . . . . . . . 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 30815	. . . . . . . 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 2764	. . . . . . . . . 10 |- F/_ y pred ( z , A , R )
51	50, 25	nfss 3596	. . . . . . . . 9 |- F/ y pred ( z , A , R ) C_ B
52	51	nf5ri 2065	. . . . . . . 8 |- ( pred ( z , A , R ) C_ B -> A. y pred ( z , A , R ) C_ B )
53	49, 52	bnj1397 30905	. . . . . . 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 477	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> z e. B )
55	3	bnj1138 30859	. . . . . . . 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 208	. . . . . . 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 827	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> pred ( z , A , R ) C_ B )
58	57	ralrimiva 2966	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> A. z e. B pred ( z , A , R ) C_ B )
59		df-bnj19 30763	. . . . 5 |- ( TrFo ( B , A , R ) <-> A. z e. B pred ( z , A , R ) C_ B )
60	58, 59	sylibr 224	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> TrFo ( B , A , R ) )
61	3	bnj931 30841	. . . . 5 |- pred ( X , A , R ) C_ B
62	61	a1i 11	. . . 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 1246	. . 3 |- ( ( R FrSe A /\ X e. A ) -> ta )
65	1, 63	bnj1124 31056	. . 3 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )
66	2, 64, 65	syl2anc 693	. 2 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) C_ B )
67		bnj906 31000	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
68		iunss1 4532	. . . . 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 3784	. . . . 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 3646	. . 3 |- ( ( R FrSe A /\ X e. A ) -> B C_ C )
73		biid 251	. . . 4 |- ( ( R FrSe A /\ X e. A ) <-> ( R FrSe A /\ X e. A ) )
74		biid 251	. . . 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 31065	. . 3 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = C )
76	72, 75	sseqtr4d 3642	. 2 |- ( ( R FrSe A /\ X e. A ) -> B C_ trCl ( X , A , R ) )
77	66, 76	eqssd 3620	1 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = B )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   C_ wss 3574   U_ciun 4520   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760   TrFow-bnj19 30762

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1414  31105