Step |
Hyp |
Ref |
Expression |
1 |
|
ovexd |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 } ↑m 𝐵 ) ∈ V ) |
2 |
|
snex |
⊢ { 𝐴 } ∈ V |
3 |
2
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → { 𝐴 } ∈ V ) |
4 |
|
simpl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → 𝐴 ∈ 𝑉 ) |
5 |
4
|
a1d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) → 𝐴 ∈ 𝑉 ) ) |
6 |
2
|
a1i |
⊢ ( 𝐴 ∈ 𝑉 → { 𝐴 } ∈ V ) |
7 |
6
|
anim1ci |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 ∈ 𝑊 ∧ { 𝐴 } ∈ V ) ) |
8 |
|
xpexg |
⊢ ( ( 𝐵 ∈ 𝑊 ∧ { 𝐴 } ∈ V ) → ( 𝐵 × { 𝐴 } ) ∈ V ) |
9 |
7 8
|
syl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐵 × { 𝐴 } ) ∈ V ) |
10 |
9
|
a1d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑦 ∈ { 𝐴 } → ( 𝐵 × { 𝐴 } ) ∈ V ) ) |
11 |
|
velsn |
⊢ ( 𝑦 ∈ { 𝐴 } ↔ 𝑦 = 𝐴 ) |
12 |
11
|
a1i |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑦 ∈ { 𝐴 } ↔ 𝑦 = 𝐴 ) ) |
13 |
|
elmapg |
⊢ ( ( { 𝐴 } ∈ V ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ↔ 𝑥 : 𝐵 ⟶ { 𝐴 } ) ) |
14 |
6 13
|
sylan |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ↔ 𝑥 : 𝐵 ⟶ { 𝐴 } ) ) |
15 |
|
fconst2g |
⊢ ( 𝐴 ∈ 𝑉 → ( 𝑥 : 𝐵 ⟶ { 𝐴 } ↔ 𝑥 = ( 𝐵 × { 𝐴 } ) ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 : 𝐵 ⟶ { 𝐴 } ↔ 𝑥 = ( 𝐵 × { 𝐴 } ) ) ) |
17 |
14 16
|
bitr2d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝑥 = ( 𝐵 × { 𝐴 } ) ↔ 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ) ) |
18 |
12 17
|
anbi12d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝑦 ∈ { 𝐴 } ∧ 𝑥 = ( 𝐵 × { 𝐴 } ) ) ↔ ( 𝑦 = 𝐴 ∧ 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ) ) ) |
19 |
|
ancom |
⊢ ( ( 𝑦 = 𝐴 ∧ 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ) ↔ ( 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ∧ 𝑦 = 𝐴 ) ) |
20 |
18 19
|
bitr2di |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( ( 𝑥 ∈ ( { 𝐴 } ↑m 𝐵 ) ∧ 𝑦 = 𝐴 ) ↔ ( 𝑦 ∈ { 𝐴 } ∧ 𝑥 = ( 𝐵 × { 𝐴 } ) ) ) ) |
21 |
1 3 5 10 20
|
en2d |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( { 𝐴 } ↑m 𝐵 ) ≈ { 𝐴 } ) |