Theorem f1ocnvdm

Description: The value of the converse of a one-to-one onto function belongs to its domain. (Contributed by NM, 26-May-2006)

	Ref	Expression
Assertion	f1ocnvdm	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝐶 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		f1ocnv	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐴 )
2		f1of	⊢ ( ◡ 𝐹 : 𝐵 –1-1-onto→ 𝐴 → ◡ 𝐹 : 𝐵 ⟶ 𝐴 )
3	1 2	syl	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ◡ 𝐹 : 𝐵 ⟶ 𝐴 )
4	3	ffvelcdmda	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐶 ∈ 𝐵 ) → ( ◡ 𝐹 ‘ 𝐶 ) ∈ 𝐴 )