Theorem bnj882 32274

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 32038	. 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 4284	. . 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 4284	. . . 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 697	. . . . . . . . . . . . 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 694	. . . . . . . . . . . 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 969	. . . . . . . . . . . 12 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ( ph /\ ps ) ) )
11		3anass 969	. . . . . . . . . . . 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 277	. . . . . . . . . . 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 2872	. . . . . . . . . 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 2588	. . . . . . . . 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 2483	. . . . . . . 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 2532	. . . . . . 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 695	. . . . . 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 2859	. . . . 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 2588	. . . 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 2486	. . 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 2486	. 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 2486	1 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   {cab 2439   A.wral 2799   E.wrex 2800   \ cdif 3436   (/)c0 3748   {csn 3988   U_ciun 4282   suc csuc 4832   dom cdm 4951   Fn wfn 5524   ` cfv 5529   omcom 6589   predc-bnj14 32031   trClc-bnj18 32037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-rex 2805  df-iun 4284  df-bnj18 32038

This theorem is referenced by:  bnj893  32276  bnj906  32278  bnj916  32281  bnj983  32299  bnj1014  32308  bnj1145  32339  bnj1318  32371