Theorem bnj1318 31093

Description: Technical lemma for bnj60 31130. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1318	|- ( X = Y -> trCl ( X , A , R ) = trCl ( Y , A , R ) )

Proof of Theorem bnj1318

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

Step	Hyp	Ref	Expression
1		bnj602 30985	. . . . . . 7 |- ( X = Y -> pred ( X , A , R ) = pred ( Y , A , R ) )
2	1	eqeq2d 2632	. . . . . 6 |- ( X = Y -> ( ( f ` (/) ) = pred ( X , A , R ) <-> ( f ` (/) ) = pred ( Y , A , R ) ) )
3	2	3anbi2d 1404	. . . . 5 |- ( 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	. . . 4 |- ( 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	. . 3 |- ( 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		hbab1 2611	. . . 4 |- ( z 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 ) ) ) } -> A. f z 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 ) ) ) } )
7		hbab1 2611	. . . 4 |- ( z 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 ) ) ) } -> A. f z 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 ) ) ) } )
8	6, 7	bnj1316 30891	. . 3 |- ( { 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 ) ) ) } -> 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 ) )
9	5, 8	syl 17	. 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 ) )
10		biid 251	. . 3 |- ( ( f ` (/) ) = pred ( X , A , R ) <-> ( f ` (/) ) = pred ( X , A , R ) )
11		biid 251	. . 3 |- ( A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) <-> A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) )
12		eqid 2622	. . 3 |- ( om \ { (/) } ) = ( om \ { (/) } )
13		eqid 2622	. . 3 |- { 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 ( X , A , R ) /\ A. i e. om ( suc i e. n -> ( f ` suc i ) = U_ y e. ( f ` i ) pred ( y , A , R ) ) ) }
14	10, 11, 12, 13	bnj882 30996	. 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 )
15		biid 251	. . 3 |- ( ( f ` (/) ) = pred ( Y , A , R ) <-> ( f ` (/) ) = pred ( Y , A , R ) )
16		eqid 2622	. . 3 |- { 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 ) ) ) } = { 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 ) ) ) }
17	15, 11, 12, 16	bnj882 30996	. 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 )
18	9, 14, 17	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:  bnj1373  31098  bnj1408  31104  bnj1417  31109  bnj1489  31124