Metamath Proof Explorer

Theorem fnfocofob

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 is surjective. (Contributed by GL and AV, 29-Sep-2024)

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

Proof

Step Hyp Ref Expression
1 cnvimarndm 
 |-  ( `' F " ran F ) = dom F
2 fndm 
 |-  ( F Fn A -> dom F = A )
3 2 3ad2ant1 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> dom F = A )
4 1 3 eqtr2id 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> A = ( `' F " ran F ) )
5 imaeq2 
 |-  ( ran F = B -> ( `' F " ran F ) = ( `' F " B ) )
6 5 3ad2ant3 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> ( `' F " ran F ) = ( `' F " B ) )
7 4 6 eqtrd 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> A = ( `' F " B ) )
8 foeq2 
 |-  ( A = ( `' F " B ) -> ( ( G o. F ) : A -onto-> C <-> ( G o. F ) : ( `' F " B ) -onto-> C ) )
9 7 8 syl 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> ( ( G o. F ) : A -onto-> C <-> ( G o. F ) : ( `' F " B ) -onto-> C ) )
10 fnfun 
 |-  ( F Fn A -> Fun F )
11 id 
 |-  ( G : B --> C -> G : B --> C )
12 eqimss2 
 |-  ( ran F = B -> B C_ ran F )
13 funfocofob 
 |-  ( ( Fun F /\ G : B --> C /\ B C_ ran F ) -> ( ( G o. F ) : ( `' F " B ) -onto-> C <-> G : B -onto-> C ) )
14 10 11 12 13 syl3an 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> ( ( G o. F ) : ( `' F " B ) -onto-> C <-> G : B -onto-> C ) )
15 9 14 bitrd 
 |-  ( ( F Fn A /\ G : B --> C /\ ran F = B ) -> ( ( G o. F ) : A -onto-> C <-> G : B -onto-> C ) )