Step |
Hyp |
Ref |
Expression |
1 |
|
df-ac |
⊢ ( CHOICE ↔ ∀ 𝑦 ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) ) |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
|
vuniex |
⊢ ∪ 𝑥 ∈ V |
4 |
2 3
|
xpex |
⊢ ( 𝑥 × ∪ 𝑥 ) ∈ V |
5 |
|
simpl |
⊢ ( ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) → 𝑤 ∈ 𝑥 ) |
6 |
|
elunii |
⊢ ( ( 𝑣 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥 ) → 𝑣 ∈ ∪ 𝑥 ) |
7 |
6
|
ancoms |
⊢ ( ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) → 𝑣 ∈ ∪ 𝑥 ) |
8 |
5 7
|
jca |
⊢ ( ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) → ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥 ) ) |
9 |
8
|
ssopab2i |
⊢ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥 ) } |
10 |
|
df-xp |
⊢ ( 𝑥 × ∪ 𝑥 ) = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ ∪ 𝑥 ) } |
11 |
9 10
|
sseqtrri |
⊢ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ⊆ ( 𝑥 × ∪ 𝑥 ) |
12 |
4 11
|
ssexi |
⊢ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∈ V |
13 |
|
sseq2 |
⊢ ( 𝑦 = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( 𝑓 ⊆ 𝑦 ↔ 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ) |
14 |
|
dmeq |
⊢ ( 𝑦 = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → dom 𝑦 = dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) |
15 |
14
|
fneq2d |
⊢ ( 𝑦 = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( 𝑓 Fn dom 𝑦 ↔ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ) |
16 |
13 15
|
anbi12d |
⊢ ( 𝑦 = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) ↔ ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ) ) |
17 |
16
|
exbidv |
⊢ ( 𝑦 = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) ↔ ∃ 𝑓 ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ) ) |
18 |
12 17
|
spcv |
⊢ ( ∀ 𝑦 ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) → ∃ 𝑓 ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ) |
19 |
|
fndm |
⊢ ( 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → dom 𝑓 = dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) |
20 |
|
dmopab |
⊢ dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } = { 𝑤 ∣ ∃ 𝑣 ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } |
21 |
20
|
eleq2i |
⊢ ( 𝑧 ∈ dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ↔ 𝑧 ∈ { 𝑤 ∣ ∃ 𝑣 ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) |
22 |
|
vex |
⊢ 𝑧 ∈ V |
23 |
|
elequ1 |
⊢ ( 𝑤 = 𝑧 → ( 𝑤 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥 ) ) |
24 |
|
eleq2 |
⊢ ( 𝑤 = 𝑧 → ( 𝑣 ∈ 𝑤 ↔ 𝑣 ∈ 𝑧 ) ) |
25 |
23 24
|
anbi12d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧 ) ) ) |
26 |
25
|
exbidv |
⊢ ( 𝑤 = 𝑧 → ( ∃ 𝑣 ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧 ) ) ) |
27 |
22 26
|
elab |
⊢ ( 𝑧 ∈ { 𝑤 ∣ ∃ 𝑣 ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ↔ ∃ 𝑣 ( 𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧 ) ) |
28 |
|
19.42v |
⊢ ( ∃ 𝑣 ( 𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧 ) ↔ ( 𝑧 ∈ 𝑥 ∧ ∃ 𝑣 𝑣 ∈ 𝑧 ) ) |
29 |
|
n0 |
⊢ ( 𝑧 ≠ ∅ ↔ ∃ 𝑣 𝑣 ∈ 𝑧 ) |
30 |
29
|
anbi2i |
⊢ ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ↔ ( 𝑧 ∈ 𝑥 ∧ ∃ 𝑣 𝑣 ∈ 𝑧 ) ) |
31 |
28 30
|
bitr4i |
⊢ ( ∃ 𝑣 ( 𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ) |
32 |
21 27 31
|
3bitrri |
⊢ ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ↔ 𝑧 ∈ dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) |
33 |
|
eleq2 |
⊢ ( dom 𝑓 = dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( 𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ) |
34 |
32 33
|
bitr4id |
⊢ ( dom 𝑓 = dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ↔ 𝑧 ∈ dom 𝑓 ) ) |
35 |
19 34
|
syl |
⊢ ( 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ↔ 𝑧 ∈ dom 𝑓 ) ) |
36 |
35
|
adantl |
⊢ ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ↔ 𝑧 ∈ dom 𝑓 ) ) |
37 |
|
fnfun |
⊢ ( 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } → Fun 𝑓 ) |
38 |
|
funfvima3 |
⊢ ( ( Fun 𝑓 ∧ 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ( 𝑧 ∈ dom 𝑓 → ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ) ) |
39 |
38
|
ancoms |
⊢ ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ Fun 𝑓 ) → ( 𝑧 ∈ dom 𝑓 → ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ) ) |
40 |
37 39
|
sylan2 |
⊢ ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ( 𝑧 ∈ dom 𝑓 → ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ) ) |
41 |
36 40
|
sylbid |
⊢ ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ( ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) → ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ) ) |
42 |
41
|
imp |
⊢ ( ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ) → ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ) |
43 |
|
imasng |
⊢ ( 𝑧 ∈ V → ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) = { 𝑢 ∣ 𝑧 { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } 𝑢 } ) |
44 |
43
|
elv |
⊢ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) = { 𝑢 ∣ 𝑧 { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } 𝑢 } |
45 |
|
vex |
⊢ 𝑢 ∈ V |
46 |
|
elequ1 |
⊢ ( 𝑣 = 𝑢 → ( 𝑣 ∈ 𝑧 ↔ 𝑢 ∈ 𝑧 ) ) |
47 |
46
|
anbi2d |
⊢ ( 𝑣 = 𝑢 → ( ( 𝑧 ∈ 𝑥 ∧ 𝑣 ∈ 𝑧 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧 ) ) ) |
48 |
|
eqid |
⊢ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } = { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } |
49 |
22 45 25 47 48
|
brab |
⊢ ( 𝑧 { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } 𝑢 ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧 ) ) |
50 |
49
|
abbii |
⊢ { 𝑢 ∣ 𝑧 { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } 𝑢 } = { 𝑢 ∣ ( 𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧 ) } |
51 |
44 50
|
eqtri |
⊢ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) = { 𝑢 ∣ ( 𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧 ) } |
52 |
|
ibar |
⊢ ( 𝑧 ∈ 𝑥 → ( 𝑢 ∈ 𝑧 ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧 ) ) ) |
53 |
52
|
abbi2dv |
⊢ ( 𝑧 ∈ 𝑥 → 𝑧 = { 𝑢 ∣ ( 𝑧 ∈ 𝑥 ∧ 𝑢 ∈ 𝑧 ) } ) |
54 |
51 53
|
eqtr4id |
⊢ ( 𝑧 ∈ 𝑥 → ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) = 𝑧 ) |
55 |
54
|
eleq2d |
⊢ ( 𝑧 ∈ 𝑥 → ( ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ↔ ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
56 |
55
|
ad2antrl |
⊢ ( ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ) → ( ( 𝑓 ‘ 𝑧 ) ∈ ( { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } “ { 𝑧 } ) ↔ ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
57 |
42 56
|
mpbid |
⊢ ( ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) ∧ ( 𝑧 ∈ 𝑥 ∧ 𝑧 ≠ ∅ ) ) → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) |
58 |
57
|
exp32 |
⊢ ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ( 𝑧 ∈ 𝑥 → ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) ) |
59 |
58
|
ralrimiv |
⊢ ( ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
60 |
59
|
eximi |
⊢ ( ∃ 𝑓 ( 𝑓 ⊆ { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ∧ 𝑓 Fn dom { 〈 𝑤 , 𝑣 〉 ∣ ( 𝑤 ∈ 𝑥 ∧ 𝑣 ∈ 𝑤 ) } ) → ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
61 |
18 60
|
syl |
⊢ ( ∀ 𝑦 ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) → ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
62 |
61
|
alrimiv |
⊢ ( ∀ 𝑦 ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) → ∀ 𝑥 ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
63 |
|
eqid |
⊢ ( 𝑤 ∈ dom 𝑦 ↦ ( 𝑓 ‘ { 𝑢 ∣ 𝑤 𝑦 𝑢 } ) ) = ( 𝑤 ∈ dom 𝑦 ↦ ( 𝑓 ‘ { 𝑢 ∣ 𝑤 𝑦 𝑢 } ) ) |
64 |
63
|
aceq3lem |
⊢ ( ∀ 𝑥 ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) ) |
65 |
64
|
alrimiv |
⊢ ( ∀ 𝑥 ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) → ∀ 𝑦 ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) ) |
66 |
62 65
|
impbii |
⊢ ( ∀ 𝑦 ∃ 𝑓 ( 𝑓 ⊆ 𝑦 ∧ 𝑓 Fn dom 𝑦 ) ↔ ∀ 𝑥 ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |
67 |
1 66
|
bitri |
⊢ ( CHOICE ↔ ∀ 𝑥 ∃ 𝑓 ∀ 𝑧 ∈ 𝑥 ( 𝑧 ≠ ∅ → ( 𝑓 ‘ 𝑧 ) ∈ 𝑧 ) ) |