Theorem recsfval 6283

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 6273	. 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 3799	. 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 2189	1 |- recs ( F ) = U. A

Syntax hints:   /\ wa 103   = wceq 1343   {cab 2151   A.wral 2444   E.wrex 2445   U.cuni 3789   Oncon0 4341   |` cres 4606   Fn wfn 5183   ` cfv 5188  recscrecs 6272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790  df-recs 6273

This theorem is referenced by:  tfrlem6  6284  tfrlem7  6285  tfrlem8  6286  tfrlem9  6287  tfrlemibfn  6296  tfrlemiubacc  6298  tfrlemi14d  6301  tfrexlem  6302