Theorem fdmeu

Description: There is exactly one codomain element for each element of the domain of a function. (Contributed by AV, 20-Apr-2025)

	Ref	Expression
Assertion	fdmeu	|- ( ( F : A --> B /\ X e. A ) -> E! y e. B ( F ` X ) = y )

Proof

Step	Hyp	Ref	Expression
1		feu	|- ( ( F : A --> B /\ X e. A ) -> E! y e. B <. X , y >. e. F )
2		ffn	|- ( F : A --> B -> F Fn A )
3	2	anim1i	|- ( ( F : A --> B /\ X e. A ) -> ( F Fn A /\ X e. A ) )
4	3	adantr	|- ( ( ( F : A --> B /\ X e. A ) /\ y e. B ) -> ( F Fn A /\ X e. A ) )
5		fnopfvb	|- ( ( F Fn A /\ X e. A ) -> ( ( F ` X ) = y <-> <. X , y >. e. F ) )
6	4 5	syl	|- ( ( ( F : A --> B /\ X e. A ) /\ y e. B ) -> ( ( F ` X ) = y <-> <. X , y >. e. F ) )
7	6	reubidva	|- ( ( F : A --> B /\ X e. A ) -> ( E! y e. B ( F ` X ) = y <-> E! y e. B <. X , y >. e. F ) )
8	1 7	mpbird	|- ( ( F : A --> B /\ X e. A ) -> E! y e. B ( F ` X ) = y )