Theorem bnj882 29809

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 29572	. 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 4271	. . 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 4271	. . . 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 711	. . . . . . . . . . . . 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 708	. . . . . . . . . . . 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 1011	. . . . . . . . . . . 12 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ( ph /\ ps ) ) )
11		3anass 1011	. . . . . . . . . . . 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 285	. . . . . . . . . . 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 2894	. . . . . . . . . 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 2587	. . . . . . . . 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 2493	. . . . . . . 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 2541	. . . . . . 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 709	. . . . . 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 2879	. . . . 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 2587	. . . 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 2496	. . 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 2496	. 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 2496	1 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   \ cdif 3387   (/)c0 3722   {csn 3959   U_ciun 4269   dom cdm 4839   suc csuc 5432   Fn wfn 5584   ` cfv 5589   omcom 6711   predc-bnj14 29565   trClc-bnj18 29571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-rex 2762  df-iun 4271  df-bnj18 29572

This theorem is referenced by:  bnj893  29811  bnj906  29813  bnj916  29816  bnj983  29834  bnj1014  29843  bnj1145  29874  bnj1318  29906