Step |
Hyp |
Ref |
Expression |
1 |
|
df-recs |
⊢ recs ( 𝐹 ) = wrecs ( E , On , 𝐹 ) |
2 |
|
df-wrecs |
⊢ wrecs ( E , On , 𝐹 ) = frecs ( E , On , ( 𝐹 ∘ 2nd ) ) |
3 |
|
df-frecs |
⊢ frecs ( E , On , ( 𝐹 ∘ 2nd ) ) = ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) } |
4 |
|
3anass |
⊢ ( ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ( 𝑓 Fn 𝑥 ∧ ( ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ) ) |
5 |
|
vex |
⊢ 𝑥 ∈ V |
6 |
5
|
elon |
⊢ ( 𝑥 ∈ On ↔ Ord 𝑥 ) |
7 |
|
ordsson |
⊢ ( Ord 𝑥 → 𝑥 ⊆ On ) |
8 |
|
ordtr |
⊢ ( Ord 𝑥 → Tr 𝑥 ) |
9 |
7 8
|
jca |
⊢ ( Ord 𝑥 → ( 𝑥 ⊆ On ∧ Tr 𝑥 ) ) |
10 |
|
epweon |
⊢ E We On |
11 |
|
wess |
⊢ ( 𝑥 ⊆ On → ( E We On → E We 𝑥 ) ) |
12 |
10 11
|
mpi |
⊢ ( 𝑥 ⊆ On → E We 𝑥 ) |
13 |
12
|
anim1ci |
⊢ ( ( 𝑥 ⊆ On ∧ Tr 𝑥 ) → ( Tr 𝑥 ∧ E We 𝑥 ) ) |
14 |
|
df-ord |
⊢ ( Ord 𝑥 ↔ ( Tr 𝑥 ∧ E We 𝑥 ) ) |
15 |
13 14
|
sylibr |
⊢ ( ( 𝑥 ⊆ On ∧ Tr 𝑥 ) → Ord 𝑥 ) |
16 |
9 15
|
impbii |
⊢ ( Ord 𝑥 ↔ ( 𝑥 ⊆ On ∧ Tr 𝑥 ) ) |
17 |
|
dftr3 |
⊢ ( Tr 𝑥 ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ) |
18 |
|
ssel2 |
⊢ ( ( 𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ On ) |
19 |
|
predon |
⊢ ( 𝑦 ∈ On → Pred ( E , On , 𝑦 ) = 𝑦 ) |
20 |
19
|
sseq1d |
⊢ ( 𝑦 ∈ On → ( Pred ( E , On , 𝑦 ) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥 ) ) |
21 |
18 20
|
syl |
⊢ ( ( 𝑥 ⊆ On ∧ 𝑦 ∈ 𝑥 ) → ( Pred ( E , On , 𝑦 ) ⊆ 𝑥 ↔ 𝑦 ⊆ 𝑥 ) ) |
22 |
21
|
ralbidva |
⊢ ( 𝑥 ⊆ On → ( ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ↔ ∀ 𝑦 ∈ 𝑥 𝑦 ⊆ 𝑥 ) ) |
23 |
17 22
|
bitr4id |
⊢ ( 𝑥 ⊆ On → ( Tr 𝑥 ↔ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ) |
24 |
23
|
pm5.32i |
⊢ ( ( 𝑥 ⊆ On ∧ Tr 𝑥 ) ↔ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ) |
25 |
6 16 24
|
3bitri |
⊢ ( 𝑥 ∈ On ↔ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ) |
26 |
25
|
anbi1i |
⊢ ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ( ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ) |
27 |
|
onelon |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → 𝑦 ∈ On ) |
28 |
27 19
|
syl |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → Pred ( E , On , 𝑦 ) = 𝑦 ) |
29 |
28
|
reseq2d |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) = ( 𝑓 ↾ 𝑦 ) ) |
30 |
29
|
oveq2d |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ 𝑦 ) ) ) |
31 |
|
id |
⊢ ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝑥 ) |
32 |
|
vex |
⊢ 𝑓 ∈ V |
33 |
32
|
resex |
⊢ ( 𝑓 ↾ 𝑦 ) ∈ V |
34 |
33
|
a1i |
⊢ ( 𝑦 ∈ 𝑥 → ( 𝑓 ↾ 𝑦 ) ∈ V ) |
35 |
31 34
|
opco2 |
⊢ ( 𝑦 ∈ 𝑥 → ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
36 |
35
|
adantl |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ 𝑦 ) ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
37 |
30 36
|
eqtrd |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
38 |
37
|
eqeq2d |
⊢ ( ( 𝑥 ∈ On ∧ 𝑦 ∈ 𝑥 ) → ( ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ↔ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
39 |
38
|
ralbidva |
⊢ ( 𝑥 ∈ On → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
40 |
39
|
pm5.32i |
⊢ ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
41 |
26 40
|
bitr3i |
⊢ ( ( ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
42 |
41
|
anbi2i |
⊢ ( ( 𝑓 Fn 𝑥 ∧ ( ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ) ↔ ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
43 |
|
an12 |
⊢ ( ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ↔ ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
44 |
4 42 43
|
3bitri |
⊢ ( ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
45 |
44
|
exbii |
⊢ ( ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
46 |
|
df-rex |
⊢ ( ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
47 |
45 46
|
bitr4i |
⊢ ( ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) ↔ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
48 |
47
|
abbii |
⊢ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |
49 |
48
|
unieqi |
⊢ ∪ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ⊆ On ∧ ∀ 𝑦 ∈ 𝑥 Pred ( E , On , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝑦 ( 𝐹 ∘ 2nd ) ( 𝑓 ↾ Pred ( E , On , 𝑦 ) ) ) ) } = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |
50 |
3 49
|
eqtri |
⊢ frecs ( E , On , ( 𝐹 ∘ 2nd ) ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |
51 |
1 2 50
|
3eqtri |
⊢ recs ( 𝐹 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |