Theorem bnj882 29689

Description: Definition (using hypotheses for readability) of the function giving the transitive closure of X in A by R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj882.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj882.2	|- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
bnj882.3	|- D = ( om \ { (/) } )
bnj882.4	|- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }

Assertion

Ref	Expression
bnj882	|- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )

Distinct variable groups:   A, f, i, n, y   R, f, i, n, y   f, X, i, n, y

Allowed substitution hints:   ph( y, f, i, n)   ps( y, f, i, n)   B( y, f, i, n)   D( y, f, i, n)

Proof of Theorem bnj882

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bnj18 29452	. 2 |- 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 )
2		df-iun 4244	. . 3 |- U_ f e. B U_ i e. dom f ( f ` i ) = { w | E. f e. B w e. U_ i e. dom f ( f ` i ) }
3		df-iun 4244	. . . 4 |- 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 ) = { w | E. 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 ) ) ) } w e. U_ i e. dom f ( f ` i ) }
4		bnj882.4	. . . . . . . . 9 |- B = { f | E. n e. D ( f Fn n /\ ph /\ ps ) }
5		bnj882.3	. . . . . . . . . . 11 |- D = ( om \ { (/) } )
6		bnj882.1	. . . . . . . . . . . . . 14 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
7		bnj882.2	. . . . . . . . . . . . . 14 |- ( ps <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
8	6, 7	anbi12i 701	. . . . . . . . . . . . 13 |- ( ( ph /\ ps ) <-> ( ( 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 ) ) ) )
9	8	anbi2i 698	. . . . . . . . . . . 12 |- ( ( f Fn n /\ ( ph /\ ps ) ) <-> ( 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		3anass 986	. . . . . . . . . . . 12 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ( ph /\ ps ) ) )
11		3anass 986	. . . . . . . . . . . 12 |- ( ( 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 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	9, 10, 11	3bitr4i 280	. . . . . . . . . . 11 |- ( ( f Fn n /\ ph /\ ps ) <-> ( 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	5, 12	rexeqbii 2882	. . . . . . . . . 10 |- ( E. n e. D ( f Fn n /\ ph /\ ps ) <-> 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 ) ) ) )
14	13	abbii 2544	. . . . . . . . 9 |- { f | E. n e. D ( f Fn n /\ ph /\ ps ) } = { 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	4, 14	eqtri 2450	. . . . . . . 8 |- B = { 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	15	eleq2i 2498	. . . . . . 7 |- ( f e. B <-> 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 ) ) ) } )
17	16	anbi1i 699	. . . . . 6 |- ( ( f e. B /\ w e. U_ i e. dom f ( f ` i ) ) <-> ( 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 ) ) ) } /\ w e. U_ i e. dom f ( f ` i ) ) )
18	17	rexbii2 2864	. . . . 5 |- ( E. f e. B w e. U_ i e. dom f ( f ` i ) <-> E. 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 ) ) ) } w e. U_ i e. dom f ( f ` i ) )
19	18	abbii 2544	. . . 4 |- { w | E. f e. B w e. U_ i e. dom f ( f ` i ) } = { w | E. 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 ) ) ) } w e. U_ i e. dom f ( f ` i ) }
20	3, 19	eqtr4i 2453	. . 3 |- 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 ) = { w | E. f e. B w e. U_ i e. dom f ( f ` i ) }
21	2, 20	eqtr4i 2453	. 2 |- U_ f e. B U_ i e. dom f ( f ` i ) = 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 )
22	1, 21	eqtr4i 2453	1 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   /\ w3a 982   = wceq 1437   e. wcel 1872   {cab 2414   A.wral 2714   E.wrex 2715   \ cdif 3376   (/)c0 3704   {csn 3941   U_ciun 4242   dom cdm 4796   suc csuc 5387   Fn wfn 5539   ` cfv 5544   omcom 6650   predc-bnj14 29445   trClc-bnj18 29451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-rex 2720  df-iun 4244  df-bnj18 29452

This theorem is referenced by:  bnj893  29691  bnj906  29693  bnj916  29696  bnj983  29714  bnj1014  29723  bnj1145  29754  bnj1318  29786