Theorem bnj18eq1 30997

Description: Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj18eq1	|- ( X = Y -> trCl ( X , A , R ) = trCl ( Y , A , R ) )

Proof of Theorem bnj18eq1

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

Step	Hyp	Ref	Expression
1		bnj602 30985	. . . . . . . . . . 11 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )
2	1	eqeq2d 2632	. . . . . . . . . 10 |- ( X = Y -> ( ( f ` (/) ) = pred ( X , A , R ) <-> ( f ` (/) ) = pred ( Y , A , R ) ) )
3	2	3anbi2d 1404	. . . . . . . . 9 |- ( X = Y -> ( ( 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 ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
4	3	rexbidv 3052	. . . . . . . 8 |- ( X = Y -> ( 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 ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) ) )
5	4	abbidv 2741	. . . . . . 7 |- ( X = Y -> { 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 ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } )
6	5	eleq2d 2687	. . . . . 6 |- ( X = Y -> ( 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 e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } ) )
7	6	anbi1d 741	. . . . 5 |- ( X = Y -> ( ( 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 ) ) ) } /\ x e. U_ i e. dom f ( f ` i ) ) <-> ( f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } /\ x e. U_ i e. dom f ( f ` i ) ) ) )
8	7	rexbidv2 3048	. . . 4 |- ( X = Y -> ( 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 ) ) ) } x e. U_ i e. dom f ( f ` i ) <-> E. f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } x e. U_ i e. dom f ( f ` i ) ) )
9	8	abbidv 2741	. . 3 |- ( X = Y -> { x | 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 ) ) ) } x e. U_ i e. dom f ( f ` i ) } = { x | E. f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } x e. U_ i e. dom f ( f ` i ) } )
10		df-iun 4522	. . 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 ) = { x | 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 ) ) ) } x e. U_ i e. dom f ( f ` i ) }
11		df-iun 4522	. . 3 |- U_ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , 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 ) = { x | E. f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) } x e. U_ i e. dom f ( f ` i ) }
12	9, 10, 11	3eqtr4g 2681	. 2 |- ( X = Y -> 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 ) = U_ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , 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 ) )
13		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 )
14		df-bnj18 30761	. 2 |- trCl ( Y , A , R ) = U_ f e. { f | E. n e. ( om \ { (/) } ) ( f Fn n /\ ( f ` (/) ) = pred ( Y , 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 )
15	12, 13, 14	3eqtr4g 2681	1 |- ( X = Y -> trCl ( X , A , R ) = trCl ( Y , A , R ) )

Syntax hints:   -> wi 4   /\ 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-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-br 4654  df-bnj14 30755  df-bnj18 30761

This theorem is referenced by:  bnj1137  31063