Theorem bj-mptval

Description: Value of a function given in maps-to notation. (Contributed by BJ, 30-Dec-2020)

	Ref	Expression
Hypothesis	bj-mptval.nf	⊢ Ⅎ 𝑥 𝐴
Assertion	bj-mptval	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( 𝑋 ∈ 𝐴 → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑋 ) = 𝑌 ↔ 𝑋 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		bj-mptval.nf	⊢ Ⅎ 𝑥 𝐴
2	1	fnmptf	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 )
3		fnbrfvb	⊢ ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ) → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑋 ) = 𝑌 ↔ 𝑋 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑌 ) )
4	3	ex	⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) Fn 𝐴 → ( 𝑋 ∈ 𝐴 → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑋 ) = 𝑌 ↔ 𝑋 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑌 ) ) )
5	2 4	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ( 𝑋 ∈ 𝐴 → ( ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑋 ) = 𝑌 ↔ 𝑋 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑌 ) ) )