Step |
Hyp |
Ref |
Expression |
1 |
|
brdom3.2 |
⊢ 𝐵 ∈ V |
2 |
|
reldom |
⊢ Rel ≼ |
3 |
2
|
brrelex1i |
⊢ ( 𝐴 ≼ 𝐵 → 𝐴 ∈ V ) |
4 |
|
0sdomg |
⊢ ( 𝐴 ∈ V → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) ) |
5 |
3 4
|
syl |
⊢ ( 𝐴 ≼ 𝐵 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) ) |
6 |
|
df-ne |
⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ ) |
7 |
5 6
|
bitrdi |
⊢ ( 𝐴 ≼ 𝐵 → ( ∅ ≺ 𝐴 ↔ ¬ 𝐴 = ∅ ) ) |
8 |
7
|
biimpar |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ ¬ 𝐴 = ∅ ) → ∅ ≺ 𝐴 ) |
9 |
|
fodomr |
⊢ ( ( ∅ ≺ 𝐴 ∧ 𝐴 ≼ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) |
10 |
9
|
ancoms |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ ∅ ≺ 𝐴 ) → ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) |
11 |
8 10
|
syldan |
⊢ ( ( 𝐴 ≼ 𝐵 ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) |
12 |
|
pm5.6 |
⊢ ( ( ( 𝐴 ≼ 𝐵 ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) ↔ ( 𝐴 ≼ 𝐵 → ( 𝐴 = ∅ ∨ ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) ) ) |
13 |
11 12
|
mpbi |
⊢ ( 𝐴 ≼ 𝐵 → ( 𝐴 = ∅ ∨ ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) ) |
14 |
|
br0 |
⊢ ¬ 𝑥 ∅ 𝑦 |
15 |
14
|
nex |
⊢ ¬ ∃ 𝑦 𝑥 ∅ 𝑦 |
16 |
|
exmo |
⊢ ( ∃ 𝑦 𝑥 ∅ 𝑦 ∨ ∃* 𝑦 𝑥 ∅ 𝑦 ) |
17 |
15 16
|
mtpor |
⊢ ∃* 𝑦 𝑥 ∅ 𝑦 |
18 |
17
|
ax-gen |
⊢ ∀ 𝑥 ∃* 𝑦 𝑥 ∅ 𝑦 |
19 |
|
rzal |
⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 ∅ 𝑥 ) |
20 |
|
0ex |
⊢ ∅ ∈ V |
21 |
|
breq |
⊢ ( 𝑓 = ∅ → ( 𝑥 𝑓 𝑦 ↔ 𝑥 ∅ 𝑦 ) ) |
22 |
21
|
mobidv |
⊢ ( 𝑓 = ∅ → ( ∃* 𝑦 𝑥 𝑓 𝑦 ↔ ∃* 𝑦 𝑥 ∅ 𝑦 ) ) |
23 |
22
|
albidv |
⊢ ( 𝑓 = ∅ → ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ↔ ∀ 𝑥 ∃* 𝑦 𝑥 ∅ 𝑦 ) ) |
24 |
|
breq |
⊢ ( 𝑓 = ∅ → ( 𝑦 𝑓 𝑥 ↔ 𝑦 ∅ 𝑥 ) ) |
25 |
24
|
rexbidv |
⊢ ( 𝑓 = ∅ → ( ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ↔ ∃ 𝑦 ∈ 𝐵 𝑦 ∅ 𝑥 ) ) |
26 |
25
|
ralbidv |
⊢ ( 𝑓 = ∅ → ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 ∅ 𝑥 ) ) |
27 |
23 26
|
anbi12d |
⊢ ( 𝑓 = ∅ → ( ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ↔ ( ∀ 𝑥 ∃* 𝑦 𝑥 ∅ 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 ∅ 𝑥 ) ) ) |
28 |
20 27
|
spcev |
⊢ ( ( ∀ 𝑥 ∃* 𝑦 𝑥 ∅ 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 ∅ 𝑥 ) → ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
29 |
18 19 28
|
sylancr |
⊢ ( 𝐴 = ∅ → ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
30 |
|
fofun |
⊢ ( 𝑓 : 𝐵 –onto→ 𝐴 → Fun 𝑓 ) |
31 |
|
dffun6 |
⊢ ( Fun 𝑓 ↔ ( Rel 𝑓 ∧ ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ) ) |
32 |
31
|
simprbi |
⊢ ( Fun 𝑓 → ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ) |
33 |
30 32
|
syl |
⊢ ( 𝑓 : 𝐵 –onto→ 𝐴 → ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ) |
34 |
|
dffo4 |
⊢ ( 𝑓 : 𝐵 –onto→ 𝐴 ↔ ( 𝑓 : 𝐵 ⟶ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
35 |
34
|
simprbi |
⊢ ( 𝑓 : 𝐵 –onto→ 𝐴 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) |
36 |
33 35
|
jca |
⊢ ( 𝑓 : 𝐵 –onto→ 𝐴 → ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
37 |
36
|
eximi |
⊢ ( ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 → ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
38 |
29 37
|
jaoi |
⊢ ( ( 𝐴 = ∅ ∨ ∃ 𝑓 𝑓 : 𝐵 –onto→ 𝐴 ) → ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
39 |
13 38
|
syl |
⊢ ( 𝐴 ≼ 𝐵 → ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |
40 |
|
inss1 |
⊢ ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ 𝑓 |
41 |
40
|
ssbri |
⊢ ( 𝑥 ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) 𝑦 → 𝑥 𝑓 𝑦 ) |
42 |
41
|
moimi |
⊢ ( ∃* 𝑦 𝑥 𝑓 𝑦 → ∃* 𝑦 𝑥 ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) 𝑦 ) |
43 |
42
|
alimi |
⊢ ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 → ∀ 𝑥 ∃* 𝑦 𝑥 ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) 𝑦 ) |
44 |
|
relinxp |
⊢ Rel ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) |
45 |
|
dffun6 |
⊢ ( Fun ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ↔ ( Rel ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ∧ ∀ 𝑥 ∃* 𝑦 𝑥 ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) 𝑦 ) ) |
46 |
44 45
|
mpbiran |
⊢ ( Fun ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ↔ ∀ 𝑥 ∃* 𝑦 𝑥 ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) 𝑦 ) |
47 |
43 46
|
sylibr |
⊢ ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 → Fun ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ) |
48 |
47
|
funfnd |
⊢ ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 → ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) Fn dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ) |
49 |
|
rninxp |
⊢ ( ran ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) = 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) |
50 |
49
|
biimpri |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 → ran ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) = 𝐴 ) |
51 |
48 50
|
anim12i |
⊢ ( ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) → ( ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) Fn dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ∧ ran ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) = 𝐴 ) ) |
52 |
|
df-fo |
⊢ ( ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) : dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) –onto→ 𝐴 ↔ ( ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) Fn dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ∧ ran ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) = 𝐴 ) ) |
53 |
51 52
|
sylibr |
⊢ ( ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) → ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) : dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) –onto→ 𝐴 ) |
54 |
|
vex |
⊢ 𝑓 ∈ V |
55 |
54
|
inex1 |
⊢ ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ∈ V |
56 |
55
|
dmex |
⊢ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ∈ V |
57 |
56
|
fodom |
⊢ ( ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) : dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) –onto→ 𝐴 → 𝐴 ≼ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ) |
58 |
|
inss2 |
⊢ ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ ( 𝐵 × 𝐴 ) |
59 |
|
dmss |
⊢ ( ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ ( 𝐵 × 𝐴 ) → dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ dom ( 𝐵 × 𝐴 ) ) |
60 |
58 59
|
ax-mp |
⊢ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ dom ( 𝐵 × 𝐴 ) |
61 |
|
dmxpss |
⊢ dom ( 𝐵 × 𝐴 ) ⊆ 𝐵 |
62 |
60 61
|
sstri |
⊢ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ 𝐵 |
63 |
|
ssdomg |
⊢ ( 𝐵 ∈ V → ( dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ⊆ 𝐵 → dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ≼ 𝐵 ) ) |
64 |
1 62 63
|
mp2 |
⊢ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ≼ 𝐵 |
65 |
|
domtr |
⊢ ( ( 𝐴 ≼ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ∧ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) ≼ 𝐵 ) → 𝐴 ≼ 𝐵 ) |
66 |
64 65
|
mpan2 |
⊢ ( 𝐴 ≼ dom ( 𝑓 ∩ ( 𝐵 × 𝐴 ) ) → 𝐴 ≼ 𝐵 ) |
67 |
53 57 66
|
3syl |
⊢ ( ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) → 𝐴 ≼ 𝐵 ) |
68 |
67
|
exlimiv |
⊢ ( ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) → 𝐴 ≼ 𝐵 ) |
69 |
39 68
|
impbii |
⊢ ( 𝐴 ≼ 𝐵 ↔ ∃ 𝑓 ( ∀ 𝑥 ∃* 𝑦 𝑥 𝑓 𝑦 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝑦 𝑓 𝑥 ) ) |