Step |
Hyp |
Ref |
Expression |
1 |
|
oldval |
⊢ ( 𝐴 ∈ On → ( O ‘ 𝐴 ) = ∪ ( M “ 𝐴 ) ) |
2 |
1
|
eleq2d |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) ↔ 𝑋 ∈ ∪ ( M “ 𝐴 ) ) ) |
3 |
|
eluni |
⊢ ( 𝑋 ∈ ∪ ( M “ 𝐴 ) ↔ ∃ 𝑦 ( 𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴 ) ) ) |
4 |
|
madef |
⊢ M : On ⟶ 𝒫 No |
5 |
|
ffn |
⊢ ( M : On ⟶ 𝒫 No → M Fn On ) |
6 |
4 5
|
ax-mp |
⊢ M Fn On |
7 |
|
onss |
⊢ ( 𝐴 ∈ On → 𝐴 ⊆ On ) |
8 |
|
fvelimab |
⊢ ( ( M Fn On ∧ 𝐴 ⊆ On ) → ( 𝑦 ∈ ( M “ 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) |
9 |
6 7 8
|
sylancr |
⊢ ( 𝐴 ∈ On → ( 𝑦 ∈ ( M “ 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) |
10 |
9
|
anbi2d |
⊢ ( 𝐴 ∈ On → ( ( 𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴 ) ) ↔ ( 𝑋 ∈ 𝑦 ∧ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) ) |
11 |
10
|
exbidv |
⊢ ( 𝐴 ∈ On → ( ∃ 𝑦 ( 𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴 ) ) ↔ ∃ 𝑦 ( 𝑋 ∈ 𝑦 ∧ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) ) |
12 |
|
fvex |
⊢ ( M ‘ 𝑏 ) ∈ V |
13 |
12
|
clel3 |
⊢ ( 𝑋 ∈ ( M ‘ 𝑏 ) ↔ ∃ 𝑦 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ) |
14 |
13
|
rexbii |
⊢ ( ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ↔ ∃ 𝑏 ∈ 𝐴 ∃ 𝑦 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ) |
15 |
|
rexcom4 |
⊢ ( ∃ 𝑏 ∈ 𝐴 ∃ 𝑦 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ↔ ∃ 𝑦 ∃ 𝑏 ∈ 𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ) |
16 |
|
eqcom |
⊢ ( 𝑦 = ( M ‘ 𝑏 ) ↔ ( M ‘ 𝑏 ) = 𝑦 ) |
17 |
16
|
anbi2ci |
⊢ ( ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ↔ ( 𝑋 ∈ 𝑦 ∧ ( M ‘ 𝑏 ) = 𝑦 ) ) |
18 |
17
|
rexbii |
⊢ ( ∃ 𝑏 ∈ 𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ↔ ∃ 𝑏 ∈ 𝐴 ( 𝑋 ∈ 𝑦 ∧ ( M ‘ 𝑏 ) = 𝑦 ) ) |
19 |
|
r19.42v |
⊢ ( ∃ 𝑏 ∈ 𝐴 ( 𝑋 ∈ 𝑦 ∧ ( M ‘ 𝑏 ) = 𝑦 ) ↔ ( 𝑋 ∈ 𝑦 ∧ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) |
20 |
18 19
|
bitri |
⊢ ( ∃ 𝑏 ∈ 𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ↔ ( 𝑋 ∈ 𝑦 ∧ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) |
21 |
20
|
exbii |
⊢ ( ∃ 𝑦 ∃ 𝑏 ∈ 𝐴 ( 𝑦 = ( M ‘ 𝑏 ) ∧ 𝑋 ∈ 𝑦 ) ↔ ∃ 𝑦 ( 𝑋 ∈ 𝑦 ∧ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ) |
22 |
14 15 21
|
3bitrri |
⊢ ( ∃ 𝑦 ( 𝑋 ∈ 𝑦 ∧ ∃ 𝑏 ∈ 𝐴 ( M ‘ 𝑏 ) = 𝑦 ) ↔ ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) |
23 |
11 22
|
bitrdi |
⊢ ( 𝐴 ∈ On → ( ∃ 𝑦 ( 𝑋 ∈ 𝑦 ∧ 𝑦 ∈ ( M “ 𝐴 ) ) ↔ ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) ) |
24 |
3 23
|
syl5bb |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ∪ ( M “ 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) ) |
25 |
2 24
|
bitrd |
⊢ ( 𝐴 ∈ On → ( 𝑋 ∈ ( O ‘ 𝐴 ) ↔ ∃ 𝑏 ∈ 𝐴 𝑋 ∈ ( M ‘ 𝑏 ) ) ) |