Theorem elpreima

Description: Membership in the preimage of a class under a function. (Contributed by Jeff Madsen, 2-Sep-2009) (Proof shortened by Sky Winter, 22-Oct-2026)

	Ref	Expression
Assertion	elpreima	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		cnvimass	⊢ ( ◡ 𝐹 “ 𝐶 ) ⊆ dom 𝐹
2	1	sseli	⊢ ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) → 𝐵 ∈ dom 𝐹 )
3		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
4	3	eleq2d	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴 ) )
5	2 4	imbitrid	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) → 𝐵 ∈ 𝐴 ) )
6	5	pm4.71rd	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) ) )
7		fvimacnv	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ↔ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) )
8	7	funfni	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ↔ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) )
9	8	pm5.32da	⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐵 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) ) )
10	6 9	bitr4d	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) ) )