Theorem tfr2a 6407

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 2205	. . . 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 6405	. . 3 |- ( A e. dom recs ( G ) -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
3		tfr.1	. . . 4 |- F = recs ( G )
4	3	dmeqi 4879	. . 3 |- dom F = dom recs ( G )
5	2, 4	eleq2s 2300	. 2 |- ( A e. dom F -> (recs ( G ) ` A ) = ( G ` (recs ( G ) |` A ) ) )
6	3	fveq1i 5577	. 2 |- ( F ` A ) = (recs ( G ) ` A )
7	3	reseq1i 4955	. . 3 |- ( F |` A ) = (recs ( G ) |` A )
8	7	fveq2i 5579	. 2 |- ( G ` ( F |` A ) ) = ( G ` (recs ( G ) |` A ) )
9	5, 6, 8	3eqtr4g 2263	1 |- ( A e. dom F -> ( F ` A ) = ( G ` ( F |` A ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   Oncon0 4410   dom cdm 4675   |` cres 4677   Fn wfn 5266   ` cfv 5271  recscrecs 6390

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-recs 6391

This theorem is referenced by:  tfr0  6409  tfri2d  6422  tfrcl  6450  tfri2  6452  frecsuclem  6492