Metamath Proof Explorer

Theorem focofob

Description: If the domain of a function G equals the range of a function F , then the composition ( G o. F ) is surjective iff G and F as function to the domain of G are both surjective. Symmetric version of fnfocofob including the fact that F is a surjection onto its range. (Contributed by GL and AV, 20-Sep-2024) (Proof shortened by AV, 29-Sep-2024)

Ref Expression
Assertion focofob 
|- ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> ( ( G o. F ) : A -onto-> D <-> ( F : A -onto-> C /\ G : C -onto-> D ) ) )

Proof

Step Hyp Ref Expression
1 ffn 
 |-  ( F : A --> B -> F Fn A )
2 fnfocofob 
 |-  ( ( F Fn A /\ G : C --> D /\ ran F = C ) -> ( ( G o. F ) : A -onto-> D <-> G : C -onto-> D ) )
3 1 2 syl3an1 
 |-  ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> ( ( G o. F ) : A -onto-> D <-> G : C -onto-> D ) )
4 dffn4 
 |-  ( F Fn A <-> F : A -onto-> ran F )
5 1 4 sylib 
 |-  ( F : A --> B -> F : A -onto-> ran F )
6 5 3ad2ant1 
 |-  ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> F : A -onto-> ran F )
7 foeq3 
 |-  ( ran F = C -> ( F : A -onto-> ran F <-> F : A -onto-> C ) )
8 7 3ad2ant3 
 |-  ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> ( F : A -onto-> ran F <-> F : A -onto-> C ) )
9 6 8 mpbid 
 |-  ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> F : A -onto-> C )
10 9 biantrurd 
 |-  ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> ( G : C -onto-> D <-> ( F : A -onto-> C /\ G : C -onto-> D ) ) )
11 3 10 bitrd 
 |-  ( ( F : A --> B /\ G : C --> D /\ ran F = C ) -> ( ( G o. F ) : A -onto-> D <-> ( F : A -onto-> C /\ G : C -onto-> D ) ) )