Theorem bnj906 31000

Description: Property of trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj906	|- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )

Proof of Theorem bnj906

Dummy variables f i n y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1onn 7719	. . . . . . . 8 |- 1o e. om
2		1n0 7575	. . . . . . . 8 |- 1o =/= (/)
3		eldifsn 4317	. . . . . . . 8 |- ( 1o e. ( om \ { (/) } ) <-> ( 1o e. om /\ 1o =/= (/) ) )
4	1, 2, 3	mpbir2an 955	. . . . . . 7 |- 1o e. ( om \ { (/) } )
5	4	ne0ii 3923	. . . . . 6 |- ( om \ { (/) } ) =/= (/)
6		biid 251	. . . . . . 7 |- ( ( f ` (/) ) = pred ( X , A , R ) <-> ( f ` (/) ) = pred ( X , A , R ) )
7		biid 251	. . . . . . 7 |- ( A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
8		eqid 2622	. . . . . . 7 |- ( om \ { (/) } ) = ( om \ { (/) } )
9	6, 7, 8	bnj852 30991	. . . . . 6 |- ( ( R FrSe A /\ X e. A ) -> A. n e. ( om \ { (/) } ) E! f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
10		r19.2z 4060	. . . . . 6 |- ( ( ( om \ { (/) } ) =/= (/) /\ A. n e. ( om \ { (/) } ) E! f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) -> E. n e. ( om \ { (/) } ) E! f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
11	5, 9, 10	sylancr 695	. . . . 5 |- ( ( R FrSe A /\ X e. A ) -> E. n e. ( om \ { (/) } ) E! f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
12		euex 2494	. . . . 5 |- ( E! f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) -> E. f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
13	11, 12	bnj31 30785	. . . 4 |- ( ( R FrSe A /\ X e. A ) -> E. n e. ( om \ { (/) } ) E. f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
14		rexcom4 3225	. . . 4 |- ( E. n e. ( om \ { (/) } ) E. f ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) <-> E. f E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
15	13, 14	sylib 208	. . 3 |- ( ( R FrSe A /\ X e. A ) -> E. f E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
16		abid 2610	. . 3 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } <-> E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) )
17	15, 16	bnj1198 30866	. 2 |- ( ( R FrSe A /\ X e. A ) -> E. f f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } )
18		simp2 1062	. . . . . . 7 |- ( ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) -> ( f ` (/) ) = pred ( X , A , R ) )
19	18	reximi 3011	. . . . . 6 |- ( E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) -> E. n e. ( om \ { (/) } ) ( f ` (/) ) = pred ( X , A , R ) )
20	16, 19	sylbi 207	. . . . 5 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> E. n e. ( om \ { (/) } ) ( f ` (/) ) = pred ( X , A , R ) )
21		df-rex 2918	. . . . . 6 |- ( E. n e. ( om \ { (/) } ) ( f ` (/) ) = pred ( X , A , R ) <-> E. n ( n e. ( om \ { (/) } ) /\ ( f ` (/) ) = pred ( X , A , R ) ) )
22		19.41v 1914	. . . . . . 7 |- ( E. n ( n e. ( om \ { (/) } ) /\ ( f ` (/) ) = pred ( X , A , R ) ) <-> ( E. n n e. ( om \ { (/) } ) /\ ( f ` (/) ) = pred ( X , A , R ) ) )
23	22	simprbi 480	. . . . . 6 |- ( E. n ( n e. ( om \ { (/) } ) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( f ` (/) ) = pred ( X , A , R ) )
24	21, 23	sylbi 207	. . . . 5 |- ( E. n e. ( om \ { (/) } ) ( f ` (/) ) = pred ( X , A , R ) -> ( f ` (/) ) = pred ( X , A , R ) )
25	20, 24	syl 17	. . . 4 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> ( f ` (/) ) = pred ( X , A , R ) )
26		eqid 2622	. . . . . . 7 |- { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } = { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) }
27	8, 26	bnj900 30999	. . . . . 6 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> (/) e. dom f )
28		fveq2 6191	. . . . . . 7 |- ( i = (/) -> ( f ` i ) = ( f ` (/) ) )
29	28	ssiun2s 4564	. . . . . 6 |- ( (/) e. dom f -> ( f ` (/) ) C_ U_ i e. dom f ( f ` i ) )
30	27, 29	syl 17	. . . . 5 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> ( f ` (/) ) C_ U_ i e. dom f ( f ` i ) )
31		ssiun2 4563	. . . . . 6 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> U_ i e. dom f ( f ` i ) C_ U_ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } U_ i e. dom f ( f ` i ) )
32	6, 7, 8, 26	bnj882 30996	. . . . . 6 |- trCl ( X , A , R ) = U_ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } U_ i e. dom f ( f ` i )
33	31, 32	syl6sseqr 3652	. . . . 5 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> U_ i e. dom f ( f ` i ) C_ trCl ( X , A , R ) )
34	30, 33	sstrd 3613	. . . 4 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> ( f ` (/) ) C_ trCl ( X , A , R ) )
35	25, 34	eqsstr3d 3640	. . 3 |- ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
36	35	exlimiv 1858	. 2 |- ( E. f f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } -> pred ( X , A , R ) C_ trCl ( X , A , R ) )
37	17, 36	syl 17	1 |- ( ( R FrSe A /\ X e. A ) -> pred ( X , A , R ) C_ trCl ( X , A , R ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   {cab 2608   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   1oc1o 7553   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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

This theorem is referenced by:  bnj1137  31063  bnj1136  31065  bnj1175  31072  bnj1177  31074  bnj1413  31103  bnj1408  31104  bnj1417  31109  bnj1442  31117  bnj1452  31120