Step |
Hyp |
Ref |
Expression |
1 |
|
fabexg.1 |
⊢ 𝐹 = { 𝑥 ∣ ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) } |
2 |
|
xpexg |
⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → ( 𝐴 × 𝐵 ) ∈ V ) |
3 |
|
pwexg |
⊢ ( ( 𝐴 × 𝐵 ) ∈ V → 𝒫 ( 𝐴 × 𝐵 ) ∈ V ) |
4 |
|
fssxp |
⊢ ( 𝑥 : 𝐴 ⟶ 𝐵 → 𝑥 ⊆ ( 𝐴 × 𝐵 ) ) |
5 |
|
velpw |
⊢ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ↔ 𝑥 ⊆ ( 𝐴 × 𝐵 ) ) |
6 |
4 5
|
sylibr |
⊢ ( 𝑥 : 𝐴 ⟶ 𝐵 → 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ) |
7 |
6
|
anim1i |
⊢ ( ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) → ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) ) |
8 |
7
|
ss2abi |
⊢ { 𝑥 ∣ ( 𝑥 : 𝐴 ⟶ 𝐵 ∧ 𝜑 ) } ⊆ { 𝑥 ∣ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } |
9 |
1 8
|
eqsstri |
⊢ 𝐹 ⊆ { 𝑥 ∣ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } |
10 |
|
ssab2 |
⊢ { 𝑥 ∣ ( 𝑥 ∈ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝜑 ) } ⊆ 𝒫 ( 𝐴 × 𝐵 ) |
11 |
9 10
|
sstri |
⊢ 𝐹 ⊆ 𝒫 ( 𝐴 × 𝐵 ) |
12 |
|
ssexg |
⊢ ( ( 𝐹 ⊆ 𝒫 ( 𝐴 × 𝐵 ) ∧ 𝒫 ( 𝐴 × 𝐵 ) ∈ V ) → 𝐹 ∈ V ) |
13 |
11 12
|
mpan |
⊢ ( 𝒫 ( 𝐴 × 𝐵 ) ∈ V → 𝐹 ∈ V ) |
14 |
2 3 13
|
3syl |
⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷 ) → 𝐹 ∈ V ) |