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 } ) ) |