| 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
|
eqabi |
⊢ ( ( 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 ) ) ) ) |