Theorem elpreima

Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009)

	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	syl5ib	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) → 𝐵 ∈ 𝐴 ) )
6		fnfun	⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 )
7		fvimacnvi	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 )
8	6 7	sylan	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 )
9	8	ex	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) )
10	5 9	jcad	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) → ( 𝐵 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) ) )
11		fvimacnv	⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ↔ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) )
12	11	funfni	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ↔ 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) )
13	12	biimpd	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 → 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) )
14	13	expimpd	⊢ ( 𝐹 Fn 𝐴 → ( ( 𝐵 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) → 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ) )
15	10 14	impbid	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ ( ◡ 𝐹 “ 𝐶 ) ↔ ( 𝐵 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝐶 ) ) )