Theorem bnj882 30996

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 30761	. 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 4522	. . 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 4522	. . . 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 733	. . . . . . . . . . . . 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 730	. . . . . . . . . . . 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 1042	. . . . . . . . . . . 12 |- ( ( f Fn n /\ ph /\ ps ) <-> ( f Fn n /\ ( ph /\ ps ) ) )
11		3anass 1042	. . . . . . . . . . . 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 292	. . . . . . . . . . 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 3054	. . . . . . . . . 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 2739	. . . . . . . . 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 2644	. . . . . . . 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 2693	. . . . . . 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 731	. . . . . 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 3039	. . . . 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 2739	. . . 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 2647	. . 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 2647	. 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 2647	1 |- trCl ( X , A , R ) = U_ f e. B U_ i e. dom f ( f ` i )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   \ cdif 3571   (/)c0 3915   {csn 4177   U_ciun 4520   dom cdm 5114   suc csuc 5725   Fn wfn 5883   ` cfv 5888   omcom 7065   predc-bnj14 30754   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-rex 2918  df-iun 4522  df-bnj18 30761

This theorem is referenced by:  bnj893  30998  bnj906  31000  bnj916  31003  bnj983  31021  bnj1014  31030  bnj1145  31061  bnj1318  31093