Theorem recsfval 6461

Description: Lemma for transfinite recursion. The definition recs is the union of all acceptable functions. (Contributed by Mario Carneiro, 9-May-2015.)

Hypothesis

Ref	Expression
tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }

Assertion

Ref	Expression
recsfval	|- recs ( F ) = U. A

Distinct variable group:   x, f, y, F

Allowed substitution hints:   A( x, y, f)

Proof of Theorem recsfval

Step	Hyp	Ref	Expression
1		df-recs 6451	. 2 |- recs ( F ) = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2		tfrlem.1	. . 3 |- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
3	2	unieqi 3898	. 2 |- U. A = U. { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
4	1, 3	eqtr4i 2253	1 |- recs ( F ) = U. A

Syntax hints:   /\ wa 104   = wceq 1395   {cab 2215   A.wral 2508   E.wrex 2509   U.cuni 3888   Oncon0 4454   |` cres 4721   Fn wfn 5313   ` cfv 5318  recscrecs 6450

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-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3889  df-recs 6451

This theorem is referenced by:  tfrlem6  6462  tfrlem7  6463  tfrlem8  6464  tfrlem9  6465  tfrlemibfn  6474  tfrlemiubacc  6476  tfrlemi14d  6479  tfrexlem  6480