Theorem tfr1

Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of TakeutiZaring p. 47. We start with an arbitrary class G , normally a function, and define a class A of all "acceptable" functions. The final function we're interested in is the union F = recs ( G ) of them. F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of F . In this first part we show that F is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994) (Revised by Mario Carneiro, 18-Jan-2015)

	Ref	Expression
Hypothesis	tfr.1	|- F = recs ( G )
Assertion	tfr1	|- F Fn On

Proof

Step	Hyp	Ref	Expression
1		tfr.1	|- F = recs ( G )
2		eqid	|- { 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 ) ) ) }
3	2	tfrlem7	|- Fun recs ( G )
4	2	tfrlem14	|- dom recs ( G ) = On
5		df-fn	|- ( recs ( G ) Fn On <-> ( Fun recs ( G ) /\ dom recs ( G ) = On ) )
6	3 4 5	mpbir2an	|- recs ( G ) Fn On
7	1	fneq1i	|- ( F Fn On <-> recs ( G ) Fn On )
8	6 7	mpbir	|- F Fn On