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Metamath Proof Explorer

Theorem f1ssf1

Description: A subset of an injective function is injective. (Contributed by AV, 20-Nov-2020)

Ref Expression
Assertion f1ssf1 
|- ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G )

Proof

Step Hyp Ref Expression
1 funssres 
 |-  ( ( Fun F /\ G C_ F ) -> ( F |` dom G ) = G )
2 funres11 
 |-  ( Fun `' F -> Fun `' ( F |` dom G ) )
3 cnveq 
 |-  ( G = ( F |` dom G ) -> `' G = `' ( F |` dom G ) )
4 3 funeqd 
 |-  ( G = ( F |` dom G ) -> ( Fun `' G <-> Fun `' ( F |` dom G ) ) )
5 2 4 imbitrrid 
 |-  ( G = ( F |` dom G ) -> ( Fun `' F -> Fun `' G ) )
6 5 eqcoms 
 |-  ( ( F |` dom G ) = G -> ( Fun `' F -> Fun `' G ) )
7 1 6 syl 
 |-  ( ( Fun F /\ G C_ F ) -> ( Fun `' F -> Fun `' G ) )
8 7 ex 
 |-  ( Fun F -> ( G C_ F -> ( Fun `' F -> Fun `' G ) ) )
9 8 com23 
 |-  ( Fun F -> ( Fun `' F -> ( G C_ F -> Fun `' G ) ) )
10 9 3imp 
 |-  ( ( Fun F /\ Fun `' F /\ G C_ F ) -> Fun `' G )