Theorem bnj1137 31063

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1137.1	|- B = ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )

Assertion

Ref	Expression
bnj1137	|- ( ( R FrSe A /\ X e. A ) -> TrFo ( B , A , R ) )

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

Allowed substitution hint:   B( y)

Proof of Theorem bnj1137

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1137.1	. . . . . 6 |- B = ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
2	1	eleq2i 2693	. . . . 5 |- ( v e. B <-> v e. ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) )
3		elun 3753	. . . . 5 |- ( v e. ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) <-> ( v e. pred ( X , A , R ) \/ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) )
4	2, 3	bitri 264	. . . 4 |- ( v e. B <-> ( v e. pred ( X , A , R ) \/ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) )
5		bnj213 30952	. . . . . . . . 9 |- pred ( X , A , R ) C_ A
6	5	sseli 3599	. . . . . . . 8 |- ( v e. pred ( X , A , R ) -> v e. A )
7		bnj906 31000	. . . . . . . . 9 |- ( ( R FrSe A /\ v e. A ) -> pred ( v , A , R ) C_ trCl ( v , A , R ) )
8	7	adantlr 751	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. A ) -> pred ( v , A , R ) C_ trCl ( v , A , R ) )
9	6, 8	sylan2 491	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. pred ( X , A , R ) ) -> pred ( v , A , R ) C_ trCl ( v , A , R ) )
10		bnj906 31000	. . . . . . . . 9 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
11	10	sselda 3603	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. pred ( X , A , R ) ) -> v e. trCl ( X , A , R ) )
12		bnj18eq1 30997	. . . . . . . . 9 |- ( y = v -> trCl ( y , A , R ) = trCl ( v , A , R ) )
13	12	ssiun2s 4564	. . . . . . . 8 |- ( v e. trCl ( X , A , R ) -> trCl ( v , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
14	11, 13	syl 17	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. pred ( X , A , R ) ) -> trCl ( v , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
15	9, 14	sstrd 3613	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. pred ( X , A , R ) ) -> pred ( v , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
16		bnj1147 31062	. . . . . . . . . . 11 |- trCl ( y , A , R ) C_ A
17	16	rgenw 2924	. . . . . . . . . 10 |- A. y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ A
18		iunss 4561	. . . . . . . . . 10 |- ( U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ A <-> A. y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ A )
19	17, 18	mpbir 221	. . . . . . . . 9 |- U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ A
20	19	sseli 3599	. . . . . . . 8 |- ( v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) -> v e. A )
21	20, 8	sylan2 491	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) -> pred ( v , A , R ) C_ trCl ( v , A , R ) )
22		bnj1125 31060	. . . . . . . . . . . 12 |- ( ( R FrSe A /\ X e. A /\ y e. trCl ( X , A , R ) ) -> trCl ( y , A , R ) C_ trCl ( X , A , R ) )
23	22	3expia 1267	. . . . . . . . . . 11 |- ( ( R FrSe A /\ X e. A ) -> ( y e. trCl ( X , A , R ) -> trCl ( y , A , R ) C_ trCl ( X , A , R ) ) )
24	23	ralrimiv 2965	. . . . . . . . . 10 |- ( ( R FrSe A /\ X e. A ) -> A. y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ trCl ( X , A , R ) )
25		iunss 4561	. . . . . . . . . 10 |- ( U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ trCl ( X , A , R ) <-> A. y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ trCl ( X , A , R ) )
26	24, 25	sylibr 224	. . . . . . . . 9 |- ( ( R FrSe A /\ X e. A ) -> U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ trCl ( X , A , R ) )
27	26	sselda 3603	. . . . . . . 8 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) -> v e. trCl ( X , A , R ) )
28	27, 13	syl 17	. . . . . . 7 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) -> trCl ( v , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
29	21, 28	sstrd 3613	. . . . . 6 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) -> pred ( v , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
30	15, 29	jaodan 826	. . . . 5 |- ( ( ( R FrSe A /\ X e. A ) /\ ( v e. pred ( X , A , R ) \/ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) ) -> pred ( v , A , R ) C_ U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
31		ssun2 3777	. . . . . 6 |- U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ ( pred ( X , A , R ) u. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) )
32	31, 1	sseqtr4i 3638	. . . . 5 |- U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) C_ B
33	30, 32	syl6ss 3615	. . . 4 |- ( ( ( R FrSe A /\ X e. A ) /\ ( v e. pred ( X , A , R ) \/ v e. U_ y e. trCl ( X , A , R ) trCl ( y , A , R ) ) ) -> pred ( v , A , R ) C_ B )
34	4, 33	sylan2b 492	. . 3 |- ( ( ( R FrSe A /\ X e. A ) /\ v e. B ) -> pred ( v , A , R ) C_ B )
35	34	ralrimiva 2966	. 2 |- ( ( R FrSe A /\ X e. A ) -> A. v e. B pred ( v , A , R ) C_ B )
36		df-bnj19 30763	. 2 |- ( TrFo ( B , A , R ) <-> A. v e. B pred ( v , A , R ) C_ B )
37	35, 36	sylibr 224	1 |- ( ( R FrSe A /\ X e. A ) -> TrFo ( B , A , R ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   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:  bnj1136  31065