Theorem tfr2a 6336

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 2187	. . . 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 6334	. . 3 |- ( A e. dom recs ( G ) -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
3		tfr.1	. . . 4 |- F = recs ( G )
4	3	dmeqi 4840	. . 3 |- dom F = dom recs ( G )
5	2, 4	eleq2s 2282	. 2 |- ( A e. dom F -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
6	3	fveq1i 5528	. 2 |- ( F ` A ) = (recs ( G ) ` A )
7	3	reseq1i 4915	. . 3 |- ( F |` A ) = (recs ( G ) |` A )
8	7	fveq2i 5530	. 2 |- ( G ` ( F |` A ) ) = ( G ` (recs ( G ) |` A ) )
9	5, 6, 8	3eqtr4g 2245	1 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   Oncon0 4375   dom cdm 4638   |` cres 4640   Fn wfn 5223   ` cfv 5228  recscrecs 6319

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-setind 4548

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-recs 6320

This theorem is referenced by:  tfr0  6338  tfri2d  6351  tfrcl  6379  tfri2  6381  frecsuclem  6421