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Metamath Proof Explorer

Theorem elmapsnd

Description: Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021)

Ref Expression
Hypotheses elmapsnd.1 ⊢ ( 𝜑 → 𝐹 Fn { 𝐴 } )
elmapsnd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
elmapsnd.3 ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝐵 )
Assertion elmapsnd ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑m { 𝐴 } ) )

Proof

Step Hyp Ref Expression
1 elmapsnd.1 ⊢ ( 𝜑 → 𝐹 Fn { 𝐴 } )
2 elmapsnd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
3 elmapsnd.3 ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝐵 )
4 elsni ⊢ ( 𝑥 ∈ { 𝐴 } → 𝑥 = 𝐴 )
5 4 fveq2d ⊢ ( 𝑥 ∈ { 𝐴 } → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐴 ) )
6 5 adantl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝐴 } ) → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝐴 ) )
7 3 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝐴 } ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝐵 )
8 6 7 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { 𝐴 } ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
9 8 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ { 𝐴 } ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
10 1 9 jca ⊢ ( 𝜑 → ( 𝐹 Fn { 𝐴 } ∧ ∀ 𝑥 ∈ { 𝐴 } ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
11 ffnfv ⊢ ( 𝐹 : { 𝐴 } ⟶ 𝐵 ↔ ( 𝐹 Fn { 𝐴 } ∧ ∀ 𝑥 ∈ { 𝐴 } ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
12 10 11 sylibr ⊢ ( 𝜑 → 𝐹 : { 𝐴 } ⟶ 𝐵 )
13 snex ⊢ { 𝐴 } ∈ V
14 13 a1i ⊢ ( 𝜑 → { 𝐴 } ∈ V )
15 2 14 elmapd ⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐵 ↑m { 𝐴 } ) ↔ 𝐹 : { 𝐴 } ⟶ 𝐵 ) )
16 12 15 mpbird ⊢ ( 𝜑 → 𝐹 ∈ ( 𝐵 ↑m { 𝐴 } ) )