Step |
Hyp |
Ref |
Expression |
1 |
|
simp3 |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 ⊆ 𝐵 ) |
2 |
|
sseqin2 |
⊢ ( 𝐶 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐶 ) = 𝐶 ) |
3 |
2
|
biimpi |
⊢ ( 𝐶 ⊆ 𝐵 → ( 𝐵 ∩ 𝐶 ) = 𝐶 ) |
4 |
3
|
eqcomd |
⊢ ( 𝐶 ⊆ 𝐵 → 𝐶 = ( 𝐵 ∩ 𝐶 ) ) |
5 |
|
k0004lem1 |
⊢ ( 𝐶 = ( 𝐵 ∩ 𝐶 ) → ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴 ⟶ 𝐶 ) ) |
6 |
1 4 5
|
3syl |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 : 𝐴 ⟶ 𝐶 ) ) |
7 |
|
simp2 |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → 𝐵 ∈ 𝑉 ) |
8 |
|
simp1 |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → 𝐴 ∈ 𝑈 ) |
9 |
7 8
|
elmapd |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → ( 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) ) |
10 |
9
|
anbi1d |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → ( ( 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ) ) |
11 |
7 1
|
ssexd |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → 𝐶 ∈ V ) |
12 |
11 8
|
elmapd |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → ( 𝐹 ∈ ( 𝐶 ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ 𝐶 ) ) |
13 |
6 10 12
|
3bitr4d |
⊢ ( ( 𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ⊆ 𝐵 ) → ( ( 𝐹 ∈ ( 𝐵 ↑m 𝐴 ) ∧ ( 𝐹 “ 𝐴 ) ⊆ 𝐶 ) ↔ 𝐹 ∈ ( 𝐶 ↑m 𝐴 ) ) ) |