Theorem fimacnv

Description: The preimage of the codomain of a function is the function's domain. (Contributed by FL, 25-Jan-2007) (Proof shortened by AV, 20-Sep-2024)

	Ref	Expression
Assertion	fimacnv	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( ◡ 𝐹 “ 𝐵 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		frn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ran 𝐹 ⊆ 𝐵 )
2		cnvimassrndm	⊢ ( ran 𝐹 ⊆ 𝐵 → ( ◡ 𝐹 “ 𝐵 ) = dom 𝐹 )
3	1 2	syl	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( ◡ 𝐹 “ 𝐵 ) = dom 𝐹 )
4		fdm	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 = 𝐴 )
5	3 4	eqtrd	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( ◡ 𝐹 “ 𝐵 ) = 𝐴 )