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Metamath Proof Explorer

Theorem f1domg

Description: The domain of a one-to-one function is dominated by its codomain. (Contributed by NM, 4-Sep-2004)

Ref Expression
Assertion f1domg 
|- ( B e. C -> ( F : A -1-1-> B -> A ~<_ B ) )

Proof

Step Hyp Ref Expression
1 f1f 
 |-  ( F : A -1-1-> B -> F : A --> B )
2 f1dmex 
 |-  ( ( F : A -1-1-> B /\ B e. C ) -> A e. _V )
3 fex 
 |-  ( ( F : A --> B /\ A e. _V ) -> F e. _V )
4 1 2 3 syl2an2r 
 |-  ( ( F : A -1-1-> B /\ B e. C ) -> F e. _V )
5 4 expcom 
 |-  ( B e. C -> ( F : A -1-1-> B -> F e. _V ) )
6 f1eq1 
 |-  ( f = F -> ( f : A -1-1-> B <-> F : A -1-1-> B ) )
7 6 spcegv 
 |-  ( F e. _V -> ( F : A -1-1-> B -> E. f f : A -1-1-> B ) )
8 5 7 syli 
 |-  ( B e. C -> ( F : A -1-1-> B -> E. f f : A -1-1-> B ) )
9 brdomg 
 |-  ( B e. C -> ( A ~<_ B <-> E. f f : A -1-1-> B ) )
10 8 9 sylibrd 
 |-  ( B e. C -> ( F : A -1-1-> B -> A ~<_ B ) )