Step |
Hyp |
Ref |
Expression |
1 |
|
dfrdg2 |
⊢ ( 𝐼 ∈ V → rec ( 𝐹 , 𝐼 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
2 |
|
iftrue |
⊢ ( 𝐼 ∈ V → if ( 𝐼 ∈ V , 𝐼 , ∅ ) = 𝐼 ) |
3 |
2
|
ifeq1d |
⊢ ( 𝐼 ∈ V → if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) |
4 |
3
|
eqeq2d |
⊢ ( 𝐼 ∈ V → ( ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ↔ ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) |
5 |
4
|
ralbidv |
⊢ ( 𝐼 ∈ V → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) |
6 |
5
|
anbi2d |
⊢ ( 𝐼 ∈ V → ( ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ↔ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) ) |
7 |
6
|
rexbidv |
⊢ ( 𝐼 ∈ V → ( ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ↔ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) ) |
8 |
7
|
abbidv |
⊢ ( 𝐼 ∈ V → { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
9 |
8
|
unieqd |
⊢ ( 𝐼 ∈ V → ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , 𝐼 , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
10 |
1 9
|
eqtr4d |
⊢ ( 𝐼 ∈ V → rec ( 𝐹 , 𝐼 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
11 |
|
0ex |
⊢ ∅ ∈ V |
12 |
|
dfrdg2 |
⊢ ( ∅ ∈ V → rec ( 𝐹 , ∅ ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
13 |
11 12
|
ax-mp |
⊢ rec ( 𝐹 , ∅ ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } |
14 |
|
rdgprc |
⊢ ( ¬ 𝐼 ∈ V → rec ( 𝐹 , 𝐼 ) = rec ( 𝐹 , ∅ ) ) |
15 |
|
iffalse |
⊢ ( ¬ 𝐼 ∈ V → if ( 𝐼 ∈ V , 𝐼 , ∅ ) = ∅ ) |
16 |
15
|
ifeq1d |
⊢ ( ¬ 𝐼 ∈ V → if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) |
17 |
16
|
eqeq2d |
⊢ ( ¬ 𝐼 ∈ V → ( ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ↔ ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) |
18 |
17
|
ralbidv |
⊢ ( ¬ 𝐼 ∈ V → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) |
19 |
18
|
anbi2d |
⊢ ( ¬ 𝐼 ∈ V → ( ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ↔ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) ) |
20 |
19
|
rexbidv |
⊢ ( ¬ 𝐼 ∈ V → ( ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ↔ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) ) ) |
21 |
20
|
abbidv |
⊢ ( ¬ 𝐼 ∈ V → { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
22 |
21
|
unieqd |
⊢ ( ¬ 𝐼 ∈ V → ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , ∅ , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
23 |
13 14 22
|
3eqtr4a |
⊢ ( ¬ 𝐼 ∈ V → rec ( 𝐹 , 𝐼 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } ) |
24 |
10 23
|
pm2.61i |
⊢ rec ( 𝐹 , 𝐼 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = if ( 𝑦 = ∅ , if ( 𝐼 ∈ V , 𝐼 , ∅ ) , if ( Lim 𝑦 , ∪ ( 𝑓 “ 𝑦 ) , ( 𝐹 ‘ ( 𝑓 ‘ ∪ 𝑦 ) ) ) ) ) } |