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 ) ) ) |
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 ) ) ) |