Metamath Proof Explorer

Theorem frrlem2

Description: Lemma for founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012)

Ref Expression
Hypothesis frrlem1.1 
|- B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) }
Assertion frrlem2 
|- ( g e. B -> Fun g )

Proof

Step Hyp Ref Expression
1 frrlem1.1 
 |-  B = { f | E. x ( f Fn x /\ ( x C_ A /\ A. y e. x Pred ( R , A , y ) C_ x ) /\ A. y e. x ( f ` y ) = ( y G ( f |` Pred ( R , A , y ) ) ) ) }
2 1 frrlem1 
 |-  B = { g | E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) }
3 2 abeq2i 
 |-  ( g e. B <-> E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) )
4 fnfun 
 |-  ( g Fn z -> Fun g )
5 4 3ad2ant1 
 |-  ( ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) -> Fun g )
6 5 exlimiv 
 |-  ( E. z ( g Fn z /\ ( z C_ A /\ A. w e. z Pred ( R , A , w ) C_ z ) /\ A. w e. z ( g ` w ) = ( w G ( g |` Pred ( R , A , w ) ) ) ) -> Fun g )
7 3 6 sylbi 
 |-  ( g e. B -> Fun g )