Theorem bnj1408 29405

Description: Technical lemma for bnj1414 29406. 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 198	. . 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 29404	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> B e. _V )
5		simplll 735	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> R FrSe A )
6		bnj213 29253	. . . . . . . . . . 11 |- pred ( X , A , R ) C_ A
7	6	sseli 3344	. . . . . . . . . 10 |- ( z e. pred ( X , A , R ) -> z e. A )
8	7	adantl 453	. . . . . . . . 9 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> z e. A )
9		bnj906 29301	. . . . . . . . 9 |- ( ( R FrSe A /\ z e. A ) -> pred ( z , A , R ) C_ trCl ( z , A , R ) )
10	5, 8, 9	syl2anc 643	. . . . . . . 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 29394	. . . . . . . . . . 11 |- ( y = z -> trCl ( y , A , R ) = trCl ( z , A , R ) )
12	11	ssiun2s 4135	. . . . . . . . . 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 3513	. . . . . . . . . . 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 3395	. . . . . . . . . 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 16	. . . . . . . . 9 |- ( z e. pred ( X , A , R ) -> trCl ( z , A , R ) C_ B )
16	15	adantl 453	. . . . . . . 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 3358	. . . . . . 7 |- ( ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) /\ z e. pred ( X , A , R ) ) -> pred ( z , A , R ) C_ B )
18		simpr 448	. . . . . . . . . . 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 29208	. . . . . . . . . 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 228	. . . . . . . . . 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 1629	. . . . . . . . . . . . 13 |- F/ y ( R FrSe A /\ X e. A )
22		nfcv 2572	. . . . . . . . . . . . . . . 16 |- F/_ y pred ( X , A , R )
23		nfiu1 4121	. . . . . . . . . . . . . . . 16 |- F/_ y U_ y e. pred ( X , A , R ) trCl ( y , A , R )
24	22, 23	nfun 3503	. . . . . . . . . . . . . . 15 |- F/_ y ( pred ( X , A , R ) u. U_ y e. pred ( X , A , R ) trCl ( y , A , R ) )
25	3, 24	nfcxfr 2569	. . . . . . . . . . . . . 14 |- F/_ y B
26	25	nfcri 2566	. . . . . . . . . . . . 13 |- F/ y z e. B
27	21, 26	nfan 1846	. . . . . . . . . . . 12 |- F/ y ( ( R FrSe A /\ X e. A ) /\ z e. B )
28	23	nfcri 2566	. . . . . . . . . . . 12 |- F/ y z e. U_ y e. pred ( X , A , R ) trCl ( y , A , R )
29	27, 28	nfan 1846	. . . . . . . . . . 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 1778	. . . . . . . . . 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 29222	. . . . . . . . 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 735	. . . . . . . . . . . . 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 978	. . . . . . . . . . . 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 29363	. . . . . . . . . . . . 13 |- trCl ( y , A , R ) C_ A
35		simp3 959	. . . . . . . . . . . . 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 29170	. . . . . . . . . . . 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 643	. . . . . . . . . . 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 958	. . . . . . . . . . . . 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 29170	. . . . . . . . . . . 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 29361	. . . . . . . . . . . 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 1184	. . . . . . . . . . 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 3358	. . . . . . . . . 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 4134	. . . . . . . . . . . 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 979	. . . . . . . . . . 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 3513	. . . . . . . . . . . 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 3395	. . . . . . . . . . 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 16	. . . . . . . . . 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 3358	. . . . . . . . 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 29113	. . . . . . . 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 2572	. . . . . . . . . 10 |- F/_ y pred ( z , A , R )
51	50, 25	nfss 3341	. . . . . . . . 9 |- F/ y pred ( z , A , R ) C_ B
52	51	nfri 1778	. . . . . . . 8 |- ( pred ( z , A , R ) C_ B -> A. y pred ( z , A , R ) C_ B )
53	49, 52	bnj1397 29206	. . . . . . 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 448	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> z e. B )
55	3	bnj1138 29159	. . . . . . . 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 189	. . . . . . 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 762	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ z e. B ) -> pred ( z , A , R ) C_ B )
58	57	ralrimiva 2789	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> A. z e. B pred ( z , A , R ) C_ B )
59		df-bnj19 29061	. . . . 5 |- ( TrFo ( B , A , R ) <-> A. z e. B pred ( z , A , R ) C_ B )
60	58, 59	sylibr 204	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> TrFo ( B , A , R ) )
61	3	bnj931 29141	. . . . 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 1138	. . 3 |- ( ( R FrSe A /\ X e. A ) -> ta )
65	1, 63	bnj1124 29357	. . 3 |- ( ( th /\ ta ) -> trCl ( X , A , R ) C_ B )
66	2, 64, 65	syl2anc 643	. 2 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) C_ B )
67		bnj906 29301	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
68		iunss1 4104	. . . . 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 3518	. . . . 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 19	. . . 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 3389	. . 3 |- ( ( R FrSe A /\ X e. A ) -> B C_ C )
73		biid 228	. . . 4 |- ( ( R FrSe A /\ X e. A ) <-> ( R FrSe A /\ X e. A ) )
74		biid 228	. . . 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 29366	. . 3 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = C )
76	72, 75	sseqtr4d 3385	. 2 |- ( ( R FrSe A /\ X e. A ) -> B C_ trCl ( X , A , R ) )
77	66, 76	eqssd 3365	1 |- ( ( R FrSe A /\ X e. A ) -> trCl ( X , A , R ) = B )

Syntax hints:   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   /\ w3a 936   = wceq 1652   e. wcel 1725   A.wral 2705   _Vcvv 2956   u. cun 3318   C_ wss 3320   U_ciun 4093   predc-bnj14 29052   FrSe w-bnj15 29056   trClc-bnj18 29058   TrFow-bnj19 29060

This theorem is referenced by:  bnj1414  29406

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-bnj17 29051  df-bnj14 29053  df-bnj13 29055  df-bnj15 29057  df-bnj18 29059  df-bnj19 29061