Theorem f1ff1 5401

Description: If a function is one-to-one from A to B and is also a function from A to C, then it is a one-to-one function from A to C. (Contributed by BJ, 4-Jul-2022.)

Assertion

Ref	Expression
f1ff1	|- ( ( F : A -1-1-> B /\ F : A --> C ) -> F : A -1-1-> C )

Proof of Theorem f1ff1

Step	Hyp	Ref	Expression
1		frn 5346	. 2 |- ( F : A --> C -> ran F C_ C )
2		f1ssr 5400	. 2 |- ( ( F : A -1-1-> B /\ ran F C_ C ) -> F : A -1-1-> C )
3	1, 2	sylan2 284	1 |- ( ( F : A -1-1-> B /\ F : A --> C ) -> F : A -1-1-> C )

Syntax hints:   -> wi 4   /\ wa 103   C_ wss 3116   ran crn 4605   -->wf 5184   -1-1->wf1 5185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116  df-f 5192  df-f1 5193

This theorem is referenced by:  f1resf1  5403  inresflem  7025