Theorem recsfval 6548

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 6538	. 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 3926	. 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 2258	1 |- recs ( F ) = U. A

Syntax hints:   /\ wa 104   = wceq 1398   {cab 2220   A.wral 2522   E.wrex 2523   U.cuni 3916   Oncon0 4486   |` cres 4753   Fn wfn 5349   ` cfv 5354  recscrecs 6537

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-uni 3917  df-recs 6538

This theorem is referenced by:  tfrlem6  6549  tfrlem7  6550  tfrlem8  6551  tfrlem9  6552  tfrlemibfn  6561  tfrlemiubacc  6563  tfrlemi14d  6566  tfrexlem  6567