Step |
Hyp |
Ref |
Expression |
1 |
|
elmptrab2.f |
⊢ 𝐹 = ( 𝑥 ∈ V ↦ { 𝑦 ∈ 𝐵 ∣ 𝜑 } ) |
2 |
|
elmptrab2.s1 |
⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝜑 ↔ 𝜓 ) ) |
3 |
|
elmptrab2.s2 |
⊢ ( 𝑥 = 𝑋 → 𝐵 = 𝐶 ) |
4 |
|
elmptrab2.ex |
⊢ 𝐵 ∈ V |
5 |
|
elmptrab2.rc |
⊢ ( 𝑌 ∈ 𝐶 → 𝑋 ∈ 𝑊 ) |
6 |
4
|
a1i |
⊢ ( 𝑥 ∈ V → 𝐵 ∈ V ) |
7 |
1 2 3 6
|
elmptrab |
⊢ ( 𝑌 ∈ ( 𝐹 ‘ 𝑋 ) ↔ ( 𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓 ) ) |
8 |
|
3simpc |
⊢ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓 ) → ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) ) |
9 |
5
|
elexd |
⊢ ( 𝑌 ∈ 𝐶 → 𝑋 ∈ V ) |
10 |
9
|
adantr |
⊢ ( ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) → 𝑋 ∈ V ) |
11 |
|
simpl |
⊢ ( ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) → 𝑌 ∈ 𝐶 ) |
12 |
|
simpr |
⊢ ( ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) → 𝜓 ) |
13 |
10 11 12
|
3jca |
⊢ ( ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) → ( 𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓 ) ) |
14 |
8 13
|
impbii |
⊢ ( ( 𝑋 ∈ V ∧ 𝑌 ∈ 𝐶 ∧ 𝜓 ) ↔ ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) ) |
15 |
7 14
|
bitri |
⊢ ( 𝑌 ∈ ( 𝐹 ‘ 𝑋 ) ↔ ( 𝑌 ∈ 𝐶 ∧ 𝜓 ) ) |