Theorem fvelima2

Description: Function value in an image. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Assertion	fvelima2	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ ( 𝐹 “ 𝐶 ) ) → ∃ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elimag	⊢ ( 𝐵 ∈ ( 𝐹 “ 𝐶 ) → ( 𝐵 ∈ ( 𝐹 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 𝑥 𝐹 𝐵 ) )
2	1	ibi	⊢ ( 𝐵 ∈ ( 𝐹 “ 𝐶 ) → ∃ 𝑥 ∈ 𝐶 𝑥 𝐹 𝐵 )
3		df-rex	⊢ ( ∃ 𝑥 ∈ 𝐶 𝑥 𝐹 𝐵 ↔ ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) )
4	2 3	sylib	⊢ ( 𝐵 ∈ ( 𝐹 “ 𝐶 ) → ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) )
5		fnbr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 𝐹 𝐵 ) → 𝑥 ∈ 𝐴 )
6	5	adantrl	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → 𝑥 ∈ 𝐴 )
7		simprl	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → 𝑥 ∈ 𝐶 )
8	6 7	elind	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) )
9		fnfun	⊢ ( 𝐹 Fn 𝐴 → Fun 𝐹 )
10		funbrfv	⊢ ( Fun 𝐹 → ( 𝑥 𝐹 𝐵 → ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )
11	10	imp	⊢ ( ( Fun 𝐹 ∧ 𝑥 𝐹 𝐵 ) → ( 𝐹 ‘ 𝑥 ) = 𝐵 )
12	9 11	sylan	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 𝐹 𝐵 ) → ( 𝐹 ‘ 𝑥 ) = 𝐵 )
13	12	adantrl	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → ( 𝐹 ‘ 𝑥 ) = 𝐵 )
14	8 13	jca	⊢ ( ( 𝐹 Fn 𝐴 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )
15	14	ex	⊢ ( 𝐹 Fn 𝐴 → ( ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝐵 ) ) )
16	15	eximdv	⊢ ( 𝐹 Fn 𝐴 → ( ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) → ∃ 𝑥 ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝐵 ) ) )
17	16	imp	⊢ ( ( 𝐹 Fn 𝐴 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → ∃ 𝑥 ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )
18		df-rex	⊢ ( ∃ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝐵 ↔ ∃ 𝑥 ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ∧ ( 𝐹 ‘ 𝑥 ) = 𝐵 ) )
19	17 18	sylibr	⊢ ( ( 𝐹 Fn 𝐴 ∧ ∃ 𝑥 ( 𝑥 ∈ 𝐶 ∧ 𝑥 𝐹 𝐵 ) ) → ∃ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝐵 )
20	4 19	sylan2	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ ( 𝐹 “ 𝐶 ) ) → ∃ 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝐵 )