Step |
Hyp |
Ref |
Expression |
1 |
|
frrlem5.1 |
⊢ 𝐵 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ 𝐴 ∧ ∀ 𝑦 ∈ 𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 𝐺 ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } |
2 |
|
frrlem5.2 |
⊢ 𝐹 = frecs ( 𝑅 , 𝐴 , 𝐺 ) |
3 |
1 2
|
frrlem5 |
⊢ 𝐹 = ∪ 𝐵 |
4 |
3
|
dmeqi |
⊢ dom 𝐹 = dom ∪ 𝐵 |
5 |
|
dmuni |
⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 |
6 |
4 5
|
eqtri |
⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 |
7 |
6
|
sseq1i |
⊢ ( dom 𝐹 ⊆ 𝐴 ↔ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ) |
8 |
|
iunss |
⊢ ( ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ↔ ∀ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ) |
9 |
7 8
|
bitri |
⊢ ( dom 𝐹 ⊆ 𝐴 ↔ ∀ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ) |
10 |
1
|
frrlem3 |
⊢ ( 𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴 ) |
11 |
9 10
|
mprgbir |
⊢ dom 𝐹 ⊆ 𝐴 |