Theorem tfrlem14

Description: Lemma for transfinite recursion. Assuming ax-rep , dom recs e.V <-> recs e. V , so since dom recs is an ordinal, it must be equal to On . (Contributed by NM, 14-Aug-1994) (Revised by Mario Carneiro, 9-May-2015)

	Ref	Expression
Hypothesis	tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
Assertion	tfrlem14	|- dom recs ( F ) = On

Proof

Step	Hyp	Ref	Expression
1		tfrlem.1	|- A = { f | E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( F ` ( f |` y ) ) ) }
2	1	tfrlem13	|- -. recs ( F ) e. _V
3	1	tfrlem7	|- Fun recs ( F )
4		funex	|- ( ( Fun recs ( F ) /\ dom recs ( F ) e. On ) -> recs ( F ) e. _V )
5	3 4	mpan	|- ( dom recs ( F ) e. On -> recs ( F ) e. _V )
6	2 5	mto	|- -. dom recs ( F ) e. On
7	1	tfrlem8	|- Ord dom recs ( F )
8		ordeleqon	|- ( Ord dom recs ( F ) <-> ( dom recs ( F ) e. On \/ dom recs ( F ) = On ) )
9	7 8	mpbi	|- ( dom recs ( F ) e. On \/ dom recs ( F ) = On )
10	6 9	mtpor	|- dom recs ( F ) = On