Theorem tfr2a 6552

Description: A weak version of transfinite recursion. (Contributed by Mario Carneiro, 24-Jun-2015.)

Hypothesis

Ref	Expression
tfr.1	|- F = recs ( G )

Assertion

Ref	Expression
tfr2a	|- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )

Proof of Theorem tfr2a

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

Step	Hyp	Ref	Expression
1		eqid 2232	. . . 4 |- { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) } = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f |` y ) ) ) }
2	1	tfrlem9 6550	. . 3 |- ( A e. dom recs ( G ) -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
3		tfr.1	. . . 4 |- F = recs ( G )
4	3	dmeqi 4957	. . 3 |- dom F = dom recs ( G )
5	2, 4	eleq2s 2327	. 2 |- ( A e. dom F -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
6	3	fveq1i 5671	. 2 |- ( F ` A ) = (recs ( G ) ` A )
7	3	reseq1i 5034	. . 3 |- ( F |` A ) = (recs ( G ) |` A )
8	7	fveq2i 5673	. 2 |- ( G ` ( F |` A ) ) = ( G ` (recs ( G ) |` A ) )
9	5, 6, 8	3eqtr4g 2290	1 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   {cab 2218   A.wral 2520   E.wrex 2521   Oncon0 4484   dom cdm 4749   |` cres 4751   Fn wfn 5347   ` cfv 5352  recscrecs 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-recs 6536

This theorem is referenced by:  tfr0  6554  tfri2d  6567  tfrcl  6595  tfri2  6597  frecsuclem  6637