Theorem tfr2a 5959

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 2081	. . . 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 5958	. . 3 |- ( A e. dom recs ( G ) -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
3		tfr.1	. . . 4 |- F = recs ( G )
4	3	dmeqi 4554	. . 3 |- dom F = dom recs ( G )
5	2, 4	eleq2s 2173	. 2 |- ( A e. dom F -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
6	3	fveq1i 5199	. 2 |- ( F ` A ) = (recs ( G ) ` A )
7	3	reseq1i 4626	. . 3 |- ( F |` A ) = (recs ( G ) |` A )
8	7	fveq2i 5201	. 2 |- ( G ` ( F |` A ) ) = ( G ` (recs ( G ) |` A ) )
9	5, 6, 8	3eqtr4g 2138	1 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   Oncon0 4118   dom cdm 4363   |` cres 4365   Fn wfn 4917   ` cfv 4922  recscrecs 5942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-res 4375  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-recs 5943

This theorem is referenced by:  tfr0  5960  tfri2d  5973  tfri2  5975