Step |
Hyp |
Ref |
Expression |
1 |
|
caovmo.2 |
⊢ 𝐵 ∈ 𝑆 |
2 |
|
caovmo.dom |
⊢ dom 𝐹 = ( 𝑆 × 𝑆 ) |
3 |
|
caovmo.3 |
⊢ ¬ ∅ ∈ 𝑆 |
4 |
|
caovmo.com |
⊢ ( 𝑥 𝐹 𝑦 ) = ( 𝑦 𝐹 𝑥 ) |
5 |
|
caovmo.ass |
⊢ ( ( 𝑥 𝐹 𝑦 ) 𝐹 𝑧 ) = ( 𝑥 𝐹 ( 𝑦 𝐹 𝑧 ) ) |
6 |
|
caovmo.id |
⊢ ( 𝑥 ∈ 𝑆 → ( 𝑥 𝐹 𝐵 ) = 𝑥 ) |
7 |
|
oveq1 |
⊢ ( 𝑢 = 𝐴 → ( 𝑢 𝐹 𝑤 ) = ( 𝐴 𝐹 𝑤 ) ) |
8 |
7
|
eqeq1d |
⊢ ( 𝑢 = 𝐴 → ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ↔ ( 𝐴 𝐹 𝑤 ) = 𝐵 ) ) |
9 |
8
|
mobidv |
⊢ ( 𝑢 = 𝐴 → ( ∃* 𝑤 ( 𝑢 𝐹 𝑤 ) = 𝐵 ↔ ∃* 𝑤 ( 𝐴 𝐹 𝑤 ) = 𝐵 ) ) |
10 |
|
oveq2 |
⊢ ( 𝑤 = 𝑣 → ( 𝑢 𝐹 𝑤 ) = ( 𝑢 𝐹 𝑣 ) ) |
11 |
10
|
eqeq1d |
⊢ ( 𝑤 = 𝑣 → ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ↔ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) ) |
12 |
11
|
mo4 |
⊢ ( ∃* 𝑤 ( 𝑢 𝐹 𝑤 ) = 𝐵 ↔ ∀ 𝑤 ∀ 𝑣 ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → 𝑤 = 𝑣 ) ) |
13 |
|
simpr |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑢 𝐹 𝑣 ) = 𝐵 ) |
14 |
13
|
oveq2d |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑤 𝐹 ( 𝑢 𝐹 𝑣 ) ) = ( 𝑤 𝐹 𝐵 ) ) |
15 |
|
simpl |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑢 𝐹 𝑤 ) = 𝐵 ) |
16 |
15
|
oveq1d |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( ( 𝑢 𝐹 𝑤 ) 𝐹 𝑣 ) = ( 𝐵 𝐹 𝑣 ) ) |
17 |
|
vex |
⊢ 𝑢 ∈ V |
18 |
|
vex |
⊢ 𝑤 ∈ V |
19 |
|
vex |
⊢ 𝑣 ∈ V |
20 |
17 18 19 5
|
caovass |
⊢ ( ( 𝑢 𝐹 𝑤 ) 𝐹 𝑣 ) = ( 𝑢 𝐹 ( 𝑤 𝐹 𝑣 ) ) |
21 |
17 18 19 4 5
|
caov12 |
⊢ ( 𝑢 𝐹 ( 𝑤 𝐹 𝑣 ) ) = ( 𝑤 𝐹 ( 𝑢 𝐹 𝑣 ) ) |
22 |
20 21
|
eqtri |
⊢ ( ( 𝑢 𝐹 𝑤 ) 𝐹 𝑣 ) = ( 𝑤 𝐹 ( 𝑢 𝐹 𝑣 ) ) |
23 |
1
|
elexi |
⊢ 𝐵 ∈ V |
24 |
23 19 4
|
caovcom |
⊢ ( 𝐵 𝐹 𝑣 ) = ( 𝑣 𝐹 𝐵 ) |
25 |
16 22 24
|
3eqtr3g |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑤 𝐹 ( 𝑢 𝐹 𝑣 ) ) = ( 𝑣 𝐹 𝐵 ) ) |
26 |
14 25
|
eqtr3d |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑤 𝐹 𝐵 ) = ( 𝑣 𝐹 𝐵 ) ) |
27 |
15 1
|
eqeltrdi |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑢 𝐹 𝑤 ) ∈ 𝑆 ) |
28 |
2 3
|
ndmovrcl |
⊢ ( ( 𝑢 𝐹 𝑤 ) ∈ 𝑆 → ( 𝑢 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) |
29 |
27 28
|
syl |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑢 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) |
30 |
|
oveq1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 𝐹 𝐵 ) = ( 𝑤 𝐹 𝐵 ) ) |
31 |
|
id |
⊢ ( 𝑥 = 𝑤 → 𝑥 = 𝑤 ) |
32 |
30 31
|
eqeq12d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑥 𝐹 𝐵 ) = 𝑥 ↔ ( 𝑤 𝐹 𝐵 ) = 𝑤 ) ) |
33 |
32 6
|
vtoclga |
⊢ ( 𝑤 ∈ 𝑆 → ( 𝑤 𝐹 𝐵 ) = 𝑤 ) |
34 |
29 33
|
simpl2im |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑤 𝐹 𝐵 ) = 𝑤 ) |
35 |
13 1
|
eqeltrdi |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑢 𝐹 𝑣 ) ∈ 𝑆 ) |
36 |
2 3
|
ndmovrcl |
⊢ ( ( 𝑢 𝐹 𝑣 ) ∈ 𝑆 → ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) ) |
37 |
35 36
|
syl |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑢 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆 ) ) |
38 |
|
oveq1 |
⊢ ( 𝑥 = 𝑣 → ( 𝑥 𝐹 𝐵 ) = ( 𝑣 𝐹 𝐵 ) ) |
39 |
|
id |
⊢ ( 𝑥 = 𝑣 → 𝑥 = 𝑣 ) |
40 |
38 39
|
eqeq12d |
⊢ ( 𝑥 = 𝑣 → ( ( 𝑥 𝐹 𝐵 ) = 𝑥 ↔ ( 𝑣 𝐹 𝐵 ) = 𝑣 ) ) |
41 |
40 6
|
vtoclga |
⊢ ( 𝑣 ∈ 𝑆 → ( 𝑣 𝐹 𝐵 ) = 𝑣 ) |
42 |
37 41
|
simpl2im |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → ( 𝑣 𝐹 𝐵 ) = 𝑣 ) |
43 |
26 34 42
|
3eqtr3d |
⊢ ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → 𝑤 = 𝑣 ) |
44 |
43
|
ax-gen |
⊢ ∀ 𝑣 ( ( ( 𝑢 𝐹 𝑤 ) = 𝐵 ∧ ( 𝑢 𝐹 𝑣 ) = 𝐵 ) → 𝑤 = 𝑣 ) |
45 |
12 44
|
mpgbir |
⊢ ∃* 𝑤 ( 𝑢 𝐹 𝑤 ) = 𝐵 |
46 |
9 45
|
vtoclg |
⊢ ( 𝐴 ∈ 𝑆 → ∃* 𝑤 ( 𝐴 𝐹 𝑤 ) = 𝐵 ) |
47 |
|
moanimv |
⊢ ( ∃* 𝑤 ( 𝐴 ∈ 𝑆 ∧ ( 𝐴 𝐹 𝑤 ) = 𝐵 ) ↔ ( 𝐴 ∈ 𝑆 → ∃* 𝑤 ( 𝐴 𝐹 𝑤 ) = 𝐵 ) ) |
48 |
46 47
|
mpbir |
⊢ ∃* 𝑤 ( 𝐴 ∈ 𝑆 ∧ ( 𝐴 𝐹 𝑤 ) = 𝐵 ) |
49 |
|
eleq1 |
⊢ ( ( 𝐴 𝐹 𝑤 ) = 𝐵 → ( ( 𝐴 𝐹 𝑤 ) ∈ 𝑆 ↔ 𝐵 ∈ 𝑆 ) ) |
50 |
1 49
|
mpbiri |
⊢ ( ( 𝐴 𝐹 𝑤 ) = 𝐵 → ( 𝐴 𝐹 𝑤 ) ∈ 𝑆 ) |
51 |
2 3
|
ndmovrcl |
⊢ ( ( 𝐴 𝐹 𝑤 ) ∈ 𝑆 → ( 𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) |
52 |
50 51
|
syl |
⊢ ( ( 𝐴 𝐹 𝑤 ) = 𝐵 → ( 𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ) ) |
53 |
52
|
simpld |
⊢ ( ( 𝐴 𝐹 𝑤 ) = 𝐵 → 𝐴 ∈ 𝑆 ) |
54 |
53
|
ancri |
⊢ ( ( 𝐴 𝐹 𝑤 ) = 𝐵 → ( 𝐴 ∈ 𝑆 ∧ ( 𝐴 𝐹 𝑤 ) = 𝐵 ) ) |
55 |
54
|
moimi |
⊢ ( ∃* 𝑤 ( 𝐴 ∈ 𝑆 ∧ ( 𝐴 𝐹 𝑤 ) = 𝐵 ) → ∃* 𝑤 ( 𝐴 𝐹 𝑤 ) = 𝐵 ) |
56 |
48 55
|
ax-mp |
⊢ ∃* 𝑤 ( 𝐴 𝐹 𝑤 ) = 𝐵 |