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Metamath Proof Explorer

Theorem dff2

Description: Alternate definition of a mapping. (Contributed by NM, 14-Nov-2007)

Ref Expression
Assertion dff2 
|- ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )

Proof

Step Hyp Ref Expression
1 ffn 
 |-  ( F : A --> B -> F Fn A )
2 fssxp 
 |-  ( F : A --> B -> F C_ ( A X. B ) )
3 1 2 jca 
 |-  ( F : A --> B -> ( F Fn A /\ F C_ ( A X. B ) ) )
4 rnss 
 |-  ( F C_ ( A X. B ) -> ran F C_ ran ( A X. B ) )
5 rnxpss 
 |-  ran ( A X. B ) C_ B
6 4 5 sstrdi 
 |-  ( F C_ ( A X. B ) -> ran F C_ B )
7 6 anim2i 
 |-  ( ( F Fn A /\ F C_ ( A X. B ) ) -> ( F Fn A /\ ran F C_ B ) )
8 df-f 
 |-  ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
9 7 8 sylibr 
 |-  ( ( F Fn A /\ F C_ ( A X. B ) ) -> F : A --> B )
10 3 9 impbii 
 |-  ( F : A --> B <-> ( F Fn A /\ F C_ ( A X. B ) ) )