Metamath Proof Explorer

Theorem fornex

Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004)

Ref Expression
Assertion fornex 
|- ( A e. C -> ( F : A -onto-> B -> B e. _V ) )

Proof

Step Hyp Ref Expression
1 fofun 
 |-  ( F : A -onto-> B -> Fun F )
2 funrnex 
 |-  ( dom F e. C -> ( Fun F -> ran F e. _V ) )
3 1 2 syl5com 
 |-  ( F : A -onto-> B -> ( dom F e. C -> ran F e. _V ) )
4 fof 
 |-  ( F : A -onto-> B -> F : A --> B )
5 4 fdmd 
 |-  ( F : A -onto-> B -> dom F = A )
6 5 eleq1d 
 |-  ( F : A -onto-> B -> ( dom F e. C <-> A e. C ) )
7 forn 
 |-  ( F : A -onto-> B -> ran F = B )
8 7 eleq1d 
 |-  ( F : A -onto-> B -> ( ran F e. _V <-> B e. _V ) )
9 3 6 8 3imtr3d 
 |-  ( F : A -onto-> B -> ( A e. C -> B e. _V ) )
10 9 com12 
 |-  ( A e. C -> ( F : A -onto-> B -> B e. _V ) )