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Metamath Proof Explorer

Theorem f1oexrnex

Description: If the range of a 1-1 onto function is a set, the function itself is a set. (Contributed by AV, 2-Jun-2019)

Ref Expression
Assertion f1oexrnex 
|- ( ( F : A -1-1-onto-> B /\ B e. V ) -> F e. _V )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( F : A -1-1-onto-> B /\ B e. V ) -> F : A -1-1-onto-> B )
2 f1ocnv 
 |-  ( F : A -1-1-onto-> B -> `' F : B -1-1-onto-> A )
3 f1of 
 |-  ( `' F : B -1-1-onto-> A -> `' F : B --> A )
4 1 2 3 3syl 
 |-  ( ( F : A -1-1-onto-> B /\ B e. V ) -> `' F : B --> A )
5 fex 
 |-  ( ( `' F : B --> A /\ B e. V ) -> `' F e. _V )
6 4 5 sylancom 
 |-  ( ( F : A -1-1-onto-> B /\ B e. V ) -> `' F e. _V )
7 f1orel 
 |-  ( F : A -1-1-onto-> B -> Rel F )
8 7 adantr 
 |-  ( ( F : A -1-1-onto-> B /\ B e. V ) -> Rel F )
9 relcnvexb 
 |-  ( Rel F -> ( F e. _V <-> `' F e. _V ) )
10 8 9 syl 
 |-  ( ( F : A -1-1-onto-> B /\ B e. V ) -> ( F e. _V <-> `' F e. _V ) )
11 6 10 mpbird 
 |-  ( ( F : A -1-1-onto-> B /\ B e. V ) -> F e. _V )