Step |
Hyp |
Ref |
Expression |
1 |
|
dfrecs3 |
⊢ recs ( 𝐹 ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |
2 |
|
elin |
⊢ ( 𝑓 ∈ ( Funs ∩ ( ◡ Domain “ On ) ) ↔ ( 𝑓 ∈ Funs ∧ 𝑓 ∈ ( ◡ Domain “ On ) ) ) |
3 |
|
vex |
⊢ 𝑓 ∈ V |
4 |
3
|
elfuns |
⊢ ( 𝑓 ∈ Funs ↔ Fun 𝑓 ) |
5 |
|
vex |
⊢ 𝑥 ∈ V |
6 |
5 3
|
brcnv |
⊢ ( 𝑥 ◡ Domain 𝑓 ↔ 𝑓 Domain 𝑥 ) |
7 |
3 5
|
brdomain |
⊢ ( 𝑓 Domain 𝑥 ↔ 𝑥 = dom 𝑓 ) |
8 |
6 7
|
bitri |
⊢ ( 𝑥 ◡ Domain 𝑓 ↔ 𝑥 = dom 𝑓 ) |
9 |
8
|
rexbii |
⊢ ( ∃ 𝑥 ∈ On 𝑥 ◡ Domain 𝑓 ↔ ∃ 𝑥 ∈ On 𝑥 = dom 𝑓 ) |
10 |
3
|
elima |
⊢ ( 𝑓 ∈ ( ◡ Domain “ On ) ↔ ∃ 𝑥 ∈ On 𝑥 ◡ Domain 𝑓 ) |
11 |
|
risset |
⊢ ( dom 𝑓 ∈ On ↔ ∃ 𝑥 ∈ On 𝑥 = dom 𝑓 ) |
12 |
9 10 11
|
3bitr4i |
⊢ ( 𝑓 ∈ ( ◡ Domain “ On ) ↔ dom 𝑓 ∈ On ) |
13 |
4 12
|
anbi12i |
⊢ ( ( 𝑓 ∈ Funs ∧ 𝑓 ∈ ( ◡ Domain “ On ) ) ↔ ( Fun 𝑓 ∧ dom 𝑓 ∈ On ) ) |
14 |
2 13
|
bitri |
⊢ ( 𝑓 ∈ ( Funs ∩ ( ◡ Domain “ On ) ) ↔ ( Fun 𝑓 ∧ dom 𝑓 ∈ On ) ) |
15 |
3
|
eldm |
⊢ ( 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ↔ ∃ 𝑦 𝑓 ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) 𝑦 ) |
16 |
|
brdif |
⊢ ( 𝑓 ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) 𝑦 ↔ ( 𝑓 ( ◡ E ∘ Domain ) 𝑦 ∧ ¬ 𝑓 Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 𝑦 ) ) |
17 |
|
vex |
⊢ 𝑦 ∈ V |
18 |
3 17
|
brco |
⊢ ( 𝑓 ( ◡ E ∘ Domain ) 𝑦 ↔ ∃ 𝑥 ( 𝑓 Domain 𝑥 ∧ 𝑥 ◡ E 𝑦 ) ) |
19 |
7
|
anbi1i |
⊢ ( ( 𝑓 Domain 𝑥 ∧ 𝑥 ◡ E 𝑦 ) ↔ ( 𝑥 = dom 𝑓 ∧ 𝑥 ◡ E 𝑦 ) ) |
20 |
19
|
exbii |
⊢ ( ∃ 𝑥 ( 𝑓 Domain 𝑥 ∧ 𝑥 ◡ E 𝑦 ) ↔ ∃ 𝑥 ( 𝑥 = dom 𝑓 ∧ 𝑥 ◡ E 𝑦 ) ) |
21 |
3
|
dmex |
⊢ dom 𝑓 ∈ V |
22 |
|
breq1 |
⊢ ( 𝑥 = dom 𝑓 → ( 𝑥 ◡ E 𝑦 ↔ dom 𝑓 ◡ E 𝑦 ) ) |
23 |
21 22
|
ceqsexv |
⊢ ( ∃ 𝑥 ( 𝑥 = dom 𝑓 ∧ 𝑥 ◡ E 𝑦 ) ↔ dom 𝑓 ◡ E 𝑦 ) |
24 |
20 23
|
bitri |
⊢ ( ∃ 𝑥 ( 𝑓 Domain 𝑥 ∧ 𝑥 ◡ E 𝑦 ) ↔ dom 𝑓 ◡ E 𝑦 ) |
25 |
21 17
|
brcnv |
⊢ ( dom 𝑓 ◡ E 𝑦 ↔ 𝑦 E dom 𝑓 ) |
26 |
21
|
epeli |
⊢ ( 𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓 ) |
27 |
25 26
|
bitri |
⊢ ( dom 𝑓 ◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓 ) |
28 |
18 24 27
|
3bitri |
⊢ ( 𝑓 ( ◡ E ∘ Domain ) 𝑦 ↔ 𝑦 ∈ dom 𝑓 ) |
29 |
|
df-br |
⊢ ( 𝑓 Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 𝑦 ↔ 〈 𝑓 , 𝑦 〉 ∈ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) |
30 |
|
opex |
⊢ 〈 𝑓 , 𝑦 〉 ∈ V |
31 |
30
|
elfix |
⊢ ( 〈 𝑓 , 𝑦 〉 ∈ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ↔ 〈 𝑓 , 𝑦 〉 ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 〈 𝑓 , 𝑦 〉 ) |
32 |
30 30
|
brco |
⊢ ( 〈 𝑓 , 𝑦 〉 ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 〈 𝑓 , 𝑦 〉 ↔ ∃ 𝑥 ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ∧ 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ) ) |
33 |
|
ancom |
⊢ ( ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ∧ 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ) ↔ ( 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ∧ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ) ) |
34 |
5 30
|
brcnv |
⊢ ( 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ↔ 〈 𝑓 , 𝑦 〉 Apply 𝑥 ) |
35 |
3 17 5
|
brapply |
⊢ ( 〈 𝑓 , 𝑦 〉 Apply 𝑥 ↔ 𝑥 = ( 𝑓 ‘ 𝑦 ) ) |
36 |
34 35
|
bitri |
⊢ ( 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ↔ 𝑥 = ( 𝑓 ‘ 𝑦 ) ) |
37 |
36
|
anbi1i |
⊢ ( ( 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ∧ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ) ↔ ( 𝑥 = ( 𝑓 ‘ 𝑦 ) ∧ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ) ) |
38 |
33 37
|
bitri |
⊢ ( ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ∧ 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ) ↔ ( 𝑥 = ( 𝑓 ‘ 𝑦 ) ∧ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ) ) |
39 |
38
|
exbii |
⊢ ( ∃ 𝑥 ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ∧ 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ) ↔ ∃ 𝑥 ( 𝑥 = ( 𝑓 ‘ 𝑦 ) ∧ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ) ) |
40 |
|
fvex |
⊢ ( 𝑓 ‘ 𝑦 ) ∈ V |
41 |
|
breq2 |
⊢ ( 𝑥 = ( 𝑓 ‘ 𝑦 ) → ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ↔ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) ( 𝑓 ‘ 𝑦 ) ) ) |
42 |
40 41
|
ceqsexv |
⊢ ( ∃ 𝑥 ( 𝑥 = ( 𝑓 ‘ 𝑦 ) ∧ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ) ↔ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) ( 𝑓 ‘ 𝑦 ) ) |
43 |
39 42
|
bitri |
⊢ ( ∃ 𝑥 ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) 𝑥 ∧ 𝑥 ◡ Apply 〈 𝑓 , 𝑦 〉 ) ↔ 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) ( 𝑓 ‘ 𝑦 ) ) |
44 |
30 40
|
brco |
⊢ ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) ( 𝑓 ‘ 𝑦 ) ↔ ∃ 𝑥 ( 〈 𝑓 , 𝑦 〉 Restrict 𝑥 ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ) |
45 |
3 17 5
|
brrestrict |
⊢ ( 〈 𝑓 , 𝑦 〉 Restrict 𝑥 ↔ 𝑥 = ( 𝑓 ↾ 𝑦 ) ) |
46 |
45
|
anbi1i |
⊢ ( ( 〈 𝑓 , 𝑦 〉 Restrict 𝑥 ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 = ( 𝑓 ↾ 𝑦 ) ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ) |
47 |
46
|
exbii |
⊢ ( ∃ 𝑥 ( 〈 𝑓 , 𝑦 〉 Restrict 𝑥 ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ↔ ∃ 𝑥 ( 𝑥 = ( 𝑓 ↾ 𝑦 ) ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ) |
48 |
3
|
resex |
⊢ ( 𝑓 ↾ 𝑦 ) ∈ V |
49 |
|
breq1 |
⊢ ( 𝑥 = ( 𝑓 ↾ 𝑦 ) → ( 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ↔ ( 𝑓 ↾ 𝑦 ) FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ) |
50 |
48 49
|
ceqsexv |
⊢ ( ∃ 𝑥 ( 𝑥 = ( 𝑓 ↾ 𝑦 ) ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑓 ↾ 𝑦 ) FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) |
51 |
47 50
|
bitri |
⊢ ( ∃ 𝑥 ( 〈 𝑓 , 𝑦 〉 Restrict 𝑥 ∧ 𝑥 FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑓 ↾ 𝑦 ) FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ) |
52 |
48 40
|
brfullfun |
⊢ ( ( 𝑓 ↾ 𝑦 ) FullFun 𝐹 ( 𝑓 ‘ 𝑦 ) ↔ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
53 |
44 51 52
|
3bitri |
⊢ ( 〈 𝑓 , 𝑦 〉 ( FullFun 𝐹 ∘ Restrict ) ( 𝑓 ‘ 𝑦 ) ↔ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
54 |
32 43 53
|
3bitri |
⊢ ( 〈 𝑓 , 𝑦 〉 ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 〈 𝑓 , 𝑦 〉 ↔ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
55 |
29 31 54
|
3bitri |
⊢ ( 𝑓 Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 𝑦 ↔ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
56 |
55
|
notbii |
⊢ ( ¬ 𝑓 Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 𝑦 ↔ ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
57 |
28 56
|
anbi12i |
⊢ ( ( 𝑓 ( ◡ E ∘ Domain ) 𝑦 ∧ ¬ 𝑓 Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) 𝑦 ) ↔ ( 𝑦 ∈ dom 𝑓 ∧ ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
58 |
16 57
|
bitri |
⊢ ( 𝑓 ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) 𝑦 ↔ ( 𝑦 ∈ dom 𝑓 ∧ ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
59 |
58
|
exbii |
⊢ ( ∃ 𝑦 𝑓 ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) 𝑦 ↔ ∃ 𝑦 ( 𝑦 ∈ dom 𝑓 ∧ ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
60 |
15 59
|
bitri |
⊢ ( 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ dom 𝑓 ∧ ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
61 |
|
df-rex |
⊢ ( ∃ 𝑦 ∈ dom 𝑓 ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ dom 𝑓 ∧ ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
62 |
|
rexnal |
⊢ ( ∃ 𝑦 ∈ dom 𝑓 ¬ ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ↔ ¬ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
63 |
60 61 62
|
3bitr2ri |
⊢ ( ¬ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ↔ 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) |
64 |
63
|
con1bii |
⊢ ( ¬ 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ↔ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) |
65 |
14 64
|
anbi12i |
⊢ ( ( 𝑓 ∈ ( Funs ∩ ( ◡ Domain “ On ) ) ∧ ¬ 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) ↔ ( ( Fun 𝑓 ∧ dom 𝑓 ∈ On ) ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
66 |
|
anass |
⊢ ( ( ( Fun 𝑓 ∧ dom 𝑓 ∈ On ) ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ↔ ( Fun 𝑓 ∧ ( dom 𝑓 ∈ On ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
67 |
65 66
|
bitri |
⊢ ( ( 𝑓 ∈ ( Funs ∩ ( ◡ Domain “ On ) ) ∧ ¬ 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) ↔ ( Fun 𝑓 ∧ ( dom 𝑓 ∈ On ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
68 |
|
eleq1 |
⊢ ( 𝑥 = dom 𝑓 → ( 𝑥 ∈ On ↔ dom 𝑓 ∈ On ) ) |
69 |
|
raleq |
⊢ ( 𝑥 = dom 𝑓 → ( ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ↔ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
70 |
68 69
|
anbi12d |
⊢ ( 𝑥 = dom 𝑓 → ( ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ↔ ( dom 𝑓 ∈ On ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
71 |
70
|
anbi2d |
⊢ ( 𝑥 = dom 𝑓 → ( ( Fun 𝑓 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ↔ ( Fun 𝑓 ∧ ( dom 𝑓 ∈ On ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) ) |
72 |
21 71
|
ceqsexv |
⊢ ( ∃ 𝑥 ( 𝑥 = dom 𝑓 ∧ ( Fun 𝑓 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) ↔ ( Fun 𝑓 ∧ ( dom 𝑓 ∈ On ∧ ∀ 𝑦 ∈ dom 𝑓 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
73 |
|
df-fn |
⊢ ( 𝑓 Fn 𝑥 ↔ ( Fun 𝑓 ∧ dom 𝑓 = 𝑥 ) ) |
74 |
|
eqcom |
⊢ ( dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓 ) |
75 |
74
|
anbi2i |
⊢ ( ( Fun 𝑓 ∧ dom 𝑓 = 𝑥 ) ↔ ( Fun 𝑓 ∧ 𝑥 = dom 𝑓 ) ) |
76 |
|
ancom |
⊢ ( ( Fun 𝑓 ∧ 𝑥 = dom 𝑓 ) ↔ ( 𝑥 = dom 𝑓 ∧ Fun 𝑓 ) ) |
77 |
73 75 76
|
3bitri |
⊢ ( 𝑓 Fn 𝑥 ↔ ( 𝑥 = dom 𝑓 ∧ Fun 𝑓 ) ) |
78 |
77
|
anbi1i |
⊢ ( ( 𝑓 Fn 𝑥 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ↔ ( ( 𝑥 = dom 𝑓 ∧ Fun 𝑓 ) ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
79 |
|
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 ) ) ) ) ) |
80 |
|
anass |
⊢ ( ( ( 𝑥 = dom 𝑓 ∧ Fun 𝑓 ) ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ↔ ( 𝑥 = dom 𝑓 ∧ ( Fun 𝑓 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) ) |
81 |
78 79 80
|
3bitr3ri |
⊢ ( ( 𝑥 = dom 𝑓 ∧ ( Fun 𝑓 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) ↔ ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
82 |
81
|
exbii |
⊢ ( ∃ 𝑥 ( 𝑥 = dom 𝑓 ∧ ( Fun 𝑓 ∧ ( 𝑥 ∈ On ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
83 |
67 72 82
|
3bitr2i |
⊢ ( ( 𝑓 ∈ ( Funs ∩ ( ◡ Domain “ On ) ) ∧ ¬ 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) ↔ ∃ 𝑥 ( 𝑥 ∈ On ∧ ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) ) |
84 |
|
eldif |
⊢ ( 𝑓 ∈ ( ( Funs ∩ ( ◡ Domain “ On ) ) ∖ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) ↔ ( 𝑓 ∈ ( Funs ∩ ( ◡ Domain “ On ) ) ∧ ¬ 𝑓 ∈ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) ) |
85 |
|
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 ) ) ) ) ) |
86 |
83 84 85
|
3bitr4i |
⊢ ( 𝑓 ∈ ( ( Funs ∩ ( ◡ Domain “ On ) ) ∖ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) ↔ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) ) |
87 |
86
|
abbi2i |
⊢ ( ( Funs ∩ ( ◡ Domain “ On ) ) ∖ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) = { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |
88 |
87
|
unieqi |
⊢ ∪ ( ( Funs ∩ ( ◡ Domain “ On ) ) ∖ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) = ∪ { 𝑓 ∣ ∃ 𝑥 ∈ On ( 𝑓 Fn 𝑥 ∧ ∀ 𝑦 ∈ 𝑥 ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝑓 ↾ 𝑦 ) ) ) } |
89 |
1 88
|
eqtr4i |
⊢ recs ( 𝐹 ) = ∪ ( ( Funs ∩ ( ◡ Domain “ On ) ) ∖ dom ( ( ◡ E ∘ Domain ) ∖ Fix ( ◡ Apply ∘ ( FullFun 𝐹 ∘ Restrict ) ) ) ) |