Step |
Hyp |
Ref |
Expression |
1 |
|
ituni.u |
⊢ 𝑈 = ( 𝑥 ∈ V ↦ ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝑥 ) ↾ ω ) ) |
2 |
|
frsuc |
⊢ ( 𝐵 ∈ ω → ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ suc 𝐵 ) = ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) ‘ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) ) |
3 |
|
fvex |
⊢ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ∈ V |
4 |
|
unieq |
⊢ ( 𝑎 = ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) → ∪ 𝑎 = ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
5 |
|
unieq |
⊢ ( 𝑦 = 𝑎 → ∪ 𝑦 = ∪ 𝑎 ) |
6 |
5
|
cbvmptv |
⊢ ( 𝑦 ∈ V ↦ ∪ 𝑦 ) = ( 𝑎 ∈ V ↦ ∪ 𝑎 ) |
7 |
3
|
uniex |
⊢ ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ∈ V |
8 |
4 6 7
|
fvmpt |
⊢ ( ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ∈ V → ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) ‘ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) = ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
9 |
3 8
|
ax-mp |
⊢ ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) ‘ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) = ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) |
10 |
2 9
|
eqtrdi |
⊢ ( 𝐵 ∈ ω → ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ suc 𝐵 ) = ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
11 |
10
|
adantl |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ ω ) → ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ suc 𝐵 ) = ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
12 |
1
|
itunifval |
⊢ ( 𝐴 ∈ V → ( 𝑈 ‘ 𝐴 ) = ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ) |
13 |
12
|
fveq1d |
⊢ ( 𝐴 ∈ V → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ suc 𝐵 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ ω ) → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ suc 𝐵 ) ) |
15 |
12
|
fveq1d |
⊢ ( 𝐴 ∈ V → ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
16 |
15
|
adantr |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ ω ) → ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
17 |
16
|
unieqd |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ ω ) → ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ∪ ( ( rec ( ( 𝑦 ∈ V ↦ ∪ 𝑦 ) , 𝐴 ) ↾ ω ) ‘ 𝐵 ) ) |
18 |
11 14 17
|
3eqtr4d |
⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ ω ) → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) ) |
19 |
|
uni0 |
⊢ ∪ ∅ = ∅ |
20 |
19
|
eqcomi |
⊢ ∅ = ∪ ∅ |
21 |
1
|
itunifn |
⊢ ( 𝐴 ∈ V → ( 𝑈 ‘ 𝐴 ) Fn ω ) |
22 |
21
|
fndmd |
⊢ ( 𝐴 ∈ V → dom ( 𝑈 ‘ 𝐴 ) = ω ) |
23 |
22
|
eleq2d |
⊢ ( 𝐴 ∈ V → ( suc 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ↔ suc 𝐵 ∈ ω ) ) |
24 |
|
peano2b |
⊢ ( 𝐵 ∈ ω ↔ suc 𝐵 ∈ ω ) |
25 |
23 24
|
bitr4di |
⊢ ( 𝐴 ∈ V → ( suc 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ↔ 𝐵 ∈ ω ) ) |
26 |
25
|
notbid |
⊢ ( 𝐴 ∈ V → ( ¬ suc 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ↔ ¬ 𝐵 ∈ ω ) ) |
27 |
26
|
biimpar |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω ) → ¬ suc 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ) |
28 |
|
ndmfv |
⊢ ( ¬ suc 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∅ ) |
29 |
27 28
|
syl |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω ) → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∅ ) |
30 |
22
|
eleq2d |
⊢ ( 𝐴 ∈ V → ( 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ↔ 𝐵 ∈ ω ) ) |
31 |
30
|
notbid |
⊢ ( 𝐴 ∈ V → ( ¬ 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ↔ ¬ 𝐵 ∈ ω ) ) |
32 |
31
|
biimpar |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω ) → ¬ 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) ) |
33 |
|
ndmfv |
⊢ ( ¬ 𝐵 ∈ dom ( 𝑈 ‘ 𝐴 ) → ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ∅ ) |
34 |
32 33
|
syl |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω ) → ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ∅ ) |
35 |
34
|
unieqd |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω ) → ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ∪ ∅ ) |
36 |
20 29 35
|
3eqtr4a |
⊢ ( ( 𝐴 ∈ V ∧ ¬ 𝐵 ∈ ω ) → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) ) |
37 |
18 36
|
pm2.61dan |
⊢ ( 𝐴 ∈ V → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) ) |
38 |
|
0fv |
⊢ ( ∅ ‘ 𝐵 ) = ∅ |
39 |
38
|
unieqi |
⊢ ∪ ( ∅ ‘ 𝐵 ) = ∪ ∅ |
40 |
|
0fv |
⊢ ( ∅ ‘ suc 𝐵 ) = ∅ |
41 |
19 39 40
|
3eqtr4ri |
⊢ ( ∅ ‘ suc 𝐵 ) = ∪ ( ∅ ‘ 𝐵 ) |
42 |
|
fvprc |
⊢ ( ¬ 𝐴 ∈ V → ( 𝑈 ‘ 𝐴 ) = ∅ ) |
43 |
42
|
fveq1d |
⊢ ( ¬ 𝐴 ∈ V → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ( ∅ ‘ suc 𝐵 ) ) |
44 |
42
|
fveq1d |
⊢ ( ¬ 𝐴 ∈ V → ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ( ∅ ‘ 𝐵 ) ) |
45 |
44
|
unieqd |
⊢ ( ¬ 𝐴 ∈ V → ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) = ∪ ( ∅ ‘ 𝐵 ) ) |
46 |
41 43 45
|
3eqtr4a |
⊢ ( ¬ 𝐴 ∈ V → ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) ) |
47 |
37 46
|
pm2.61i |
⊢ ( ( 𝑈 ‘ 𝐴 ) ‘ suc 𝐵 ) = ∪ ( ( 𝑈 ‘ 𝐴 ) ‘ 𝐵 ) |