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Metamath Proof Explorer

Theorem ffnov

Description: An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004)

Ref Expression
Assertion ffnov 
|- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )

Proof

Step Hyp Ref Expression
1 ffnfv 
 |-  ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. w e. ( A X. B ) ( F ` w ) e. C ) )
2 fveq2 
 |-  ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3 df-ov 
 |-  ( x F y ) = ( F ` <. x , y >. )
4 2 3 eqtr4di 
 |-  ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5 4 eleq1d 
 |-  ( w = <. x , y >. -> ( ( F ` w ) e. C <-> ( x F y ) e. C ) )
6 5 ralxp 
 |-  ( A. w e. ( A X. B ) ( F ` w ) e. C <-> A. x e. A A. y e. B ( x F y ) e. C )
7 6 anbi2i 
 |-  ( ( F Fn ( A X. B ) /\ A. w e. ( A X. B ) ( F ` w ) e. C ) <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )
8 1 7 bitri 
 |-  ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )