Theorem tfr2a 6465

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 2229	. . . 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 6463	. . 3 |- ( A e. dom recs ( G ) -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
3		tfr.1	. . . 4 |- F = recs ( G )
4	3	dmeqi 4923	. . 3 |- dom F = dom recs ( G )
5	2, 4	eleq2s 2324	. 2 |- ( A e. dom F -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
6	3	fveq1i 5627	. 2 |- ( F ` A ) = (recs ( G ) ` A )
7	3	reseq1i 5000	. . 3 |- ( F |` A ) = (recs ( G ) |` A )
8	7	fveq2i 5629	. 2 |- ( G ` ( F |` A ) ) = ( G ` (recs ( G ) |` A ) )
9	5, 6, 8	3eqtr4g 2287	1 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   Oncon0 4453   dom cdm 4718   |` cres 4720   Fn wfn 5312   ` cfv 5317  recscrecs 6448

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-res 4730  df-iota 5277  df-fun 5319  df-fn 5320  df-fv 5325  df-recs 6449

This theorem is referenced by:  tfr0  6467  tfri2d  6480  tfrcl  6508  tfri2  6510  frecsuclem  6550