Step |
Hyp |
Ref |
Expression |
1 |
|
df-made |
⊢ M = recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) |
2 |
|
recsfnon |
⊢ recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) Fn On |
3 |
|
fnfun |
⊢ ( recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) Fn On → Fun recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) ) |
4 |
2 3
|
ax-mp |
⊢ Fun recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) |
5 |
|
funeq |
⊢ ( M = recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) → ( Fun M ↔ Fun recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) ) ) |
6 |
4 5
|
mpbiri |
⊢ ( M = recs ( ( 𝑥 ∈ V ↦ ( |s “ ( 𝒫 ∪ ran 𝑥 × 𝒫 ∪ ran 𝑥 ) ) ) ) → Fun M ) |
7 |
1 6
|
ax-mp |
⊢ Fun M |
8 |
|
funimaexg |
⊢ ( ( Fun M ∧ 𝐴 ∈ On ) → ( M “ 𝐴 ) ∈ V ) |
9 |
7 8
|
mpan |
⊢ ( 𝐴 ∈ On → ( M “ 𝐴 ) ∈ V ) |
10 |
9
|
uniexd |
⊢ ( 𝐴 ∈ On → ∪ ( M “ 𝐴 ) ∈ V ) |
11 |
|
imaeq2 |
⊢ ( 𝑥 = 𝐴 → ( M “ 𝑥 ) = ( M “ 𝐴 ) ) |
12 |
11
|
unieqd |
⊢ ( 𝑥 = 𝐴 → ∪ ( M “ 𝑥 ) = ∪ ( M “ 𝐴 ) ) |
13 |
|
df-old |
⊢ O = ( 𝑥 ∈ On ↦ ∪ ( M “ 𝑥 ) ) |
14 |
12 13
|
fvmptg |
⊢ ( ( 𝐴 ∈ On ∧ ∪ ( M “ 𝐴 ) ∈ V ) → ( O ‘ 𝐴 ) = ∪ ( M “ 𝐴 ) ) |
15 |
10 14
|
mpdan |
⊢ ( 𝐴 ∈ On → ( O ‘ 𝐴 ) = ∪ ( M “ 𝐴 ) ) |