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 } ) )