Step |
Hyp |
Ref |
Expression |
1 |
|
disjrnmpt2.1 |
⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
2 |
|
id |
⊢ ( 𝑦 = 𝑤 → 𝑦 = 𝑤 ) |
3 |
2
|
cbvdisjv |
⊢ ( Disj 𝑦 ∈ ran 𝐹 𝑦 ↔ Disj 𝑤 ∈ ran 𝐹 𝑤 ) |
4 |
|
id |
⊢ ( 𝑤 = 𝑣 → 𝑤 = 𝑣 ) |
5 |
4
|
ndisj2 |
⊢ ( ¬ Disj 𝑤 ∈ ran 𝐹 𝑤 ↔ ∃ 𝑤 ∈ ran 𝐹 ∃ 𝑣 ∈ ran 𝐹 ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) |
6 |
5
|
biimpi |
⊢ ( ¬ Disj 𝑤 ∈ ran 𝐹 𝑤 → ∃ 𝑤 ∈ ran 𝐹 ∃ 𝑣 ∈ ran 𝐹 ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) |
7 |
3 6
|
sylnbi |
⊢ ( ¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ∃ 𝑤 ∈ ran 𝐹 ∃ 𝑣 ∈ ran 𝐹 ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) |
8 |
1
|
elrnmpt |
⊢ ( 𝑤 ∈ ran 𝐹 → ( 𝑤 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) ) |
9 |
8
|
ibi |
⊢ ( 𝑤 ∈ ran 𝐹 → ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) |
10 |
|
nfcv |
⊢ Ⅎ 𝑧 𝐵 |
11 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
12 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑧 → 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
13 |
10 11 12
|
cbvmpt |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑧 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
14 |
1 13
|
eqtri |
⊢ 𝐹 = ( 𝑧 ∈ 𝐴 ↦ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
15 |
14
|
elrnmpt |
⊢ ( 𝑣 ∈ ran 𝐹 → ( 𝑣 ∈ ran 𝐹 ↔ ∃ 𝑧 ∈ 𝐴 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
16 |
15
|
ibi |
⊢ ( 𝑣 ∈ ran 𝐹 → ∃ 𝑧 ∈ 𝐴 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
17 |
9 16
|
anim12i |
⊢ ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) → ( ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ∧ ∃ 𝑧 ∈ 𝐴 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
18 |
|
nfv |
⊢ Ⅎ 𝑧 𝑤 = 𝐵 |
19 |
11
|
nfeq2 |
⊢ Ⅎ 𝑥 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
20 |
18 19
|
reean |
⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ∧ ∃ 𝑧 ∈ 𝐴 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
21 |
17 20
|
sylibr |
⊢ ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
22 |
21
|
adantr |
⊢ ( ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
23 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
24 |
1 23
|
nfcxfr |
⊢ Ⅎ 𝑥 𝐹 |
25 |
24
|
nfrn |
⊢ Ⅎ 𝑥 ran 𝐹 |
26 |
25
|
nfcri |
⊢ Ⅎ 𝑥 𝑤 ∈ ran 𝐹 |
27 |
25
|
nfcri |
⊢ Ⅎ 𝑥 𝑣 ∈ ran 𝐹 |
28 |
26 27
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) |
29 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) |
30 |
28 29
|
nfan |
⊢ Ⅎ 𝑥 ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) |
31 |
|
simpll |
⊢ ( ( ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ 𝑥 = 𝑧 ) → 𝑤 = 𝐵 ) |
32 |
12
|
adantl |
⊢ ( ( ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ 𝑥 = 𝑧 ) → 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
33 |
|
id |
⊢ ( 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
34 |
33
|
eqcomd |
⊢ ( 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 → ⦋ 𝑧 / 𝑥 ⦌ 𝐵 = 𝑣 ) |
35 |
34
|
ad2antlr |
⊢ ( ( ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ 𝑥 = 𝑧 ) → ⦋ 𝑧 / 𝑥 ⦌ 𝐵 = 𝑣 ) |
36 |
31 32 35
|
3eqtrd |
⊢ ( ( ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ∧ 𝑥 = 𝑧 ) → 𝑤 = 𝑣 ) |
37 |
36
|
adantll |
⊢ ( ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ∧ 𝑥 = 𝑧 ) → 𝑤 = 𝑣 ) |
38 |
|
simpll |
⊢ ( ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ∧ 𝑥 = 𝑧 ) → 𝑤 ≠ 𝑣 ) |
39 |
38
|
neneqd |
⊢ ( ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) ∧ 𝑥 = 𝑧 ) → ¬ 𝑤 = 𝑣 ) |
40 |
37 39
|
pm2.65da |
⊢ ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ¬ 𝑥 = 𝑧 ) |
41 |
40
|
neqned |
⊢ ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → 𝑥 ≠ 𝑧 ) |
42 |
41
|
adantlr |
⊢ ( ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → 𝑥 ≠ 𝑧 ) |
43 |
|
id |
⊢ ( 𝑤 = 𝐵 → 𝑤 = 𝐵 ) |
44 |
43
|
eqcomd |
⊢ ( 𝑤 = 𝐵 → 𝐵 = 𝑤 ) |
45 |
44
|
ad2antrl |
⊢ ( ( ( 𝑤 ∩ 𝑣 ) ≠ ∅ ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → 𝐵 = 𝑤 ) |
46 |
34
|
ad2antll |
⊢ ( ( ( 𝑤 ∩ 𝑣 ) ≠ ∅ ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ⦋ 𝑧 / 𝑥 ⦌ 𝐵 = 𝑣 ) |
47 |
45 46
|
ineq12d |
⊢ ( ( ( 𝑤 ∩ 𝑣 ) ≠ ∅ ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) = ( 𝑤 ∩ 𝑣 ) ) |
48 |
|
simpl |
⊢ ( ( ( 𝑤 ∩ 𝑣 ) ≠ ∅ ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) |
49 |
47 48
|
eqnetrd |
⊢ ( ( ( 𝑤 ∩ 𝑣 ) ≠ ∅ ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
50 |
49
|
adantll |
⊢ ( ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
51 |
42 50
|
jca |
⊢ ( ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ∧ ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) → ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
52 |
51
|
ex |
⊢ ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) → ( ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) → ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
53 |
52
|
adantl |
⊢ ( ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) → ( ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) → ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
54 |
53
|
reximdv |
⊢ ( ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) → ( ∃ 𝑧 ∈ 𝐴 ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) → ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
55 |
54
|
a1d |
⊢ ( ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) → ( 𝑥 ∈ 𝐴 → ( ∃ 𝑧 ∈ 𝐴 ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) → ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) ) |
56 |
30 55
|
reximdai |
⊢ ( ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) → ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑤 = 𝐵 ∧ 𝑣 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
57 |
22 56
|
mpd |
⊢ ( ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) ∧ ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
58 |
57
|
ex |
⊢ ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) → ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
59 |
58
|
a1i |
⊢ ( ¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ( ( 𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹 ) → ( ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) ) |
60 |
59
|
rexlimdvv |
⊢ ( ¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ( ∃ 𝑤 ∈ ran 𝐹 ∃ 𝑣 ∈ ran 𝐹 ( 𝑤 ≠ 𝑣 ∧ ( 𝑤 ∩ 𝑣 ) ≠ ∅ ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
61 |
7 60
|
mpd |
⊢ ( ¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
62 |
|
csbeq1 |
⊢ ( 𝑢 = 𝑧 → ⦋ 𝑢 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
63 |
62
|
ndisj2 |
⊢ ( ¬ Disj 𝑢 ∈ 𝐴 ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ↔ ∃ 𝑢 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑢 ≠ 𝑧 ∧ ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
64 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
65 |
|
nfv |
⊢ Ⅎ 𝑥 𝑢 ≠ 𝑧 |
66 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑢 / 𝑥 ⦌ 𝐵 |
67 |
66 11
|
nfin |
⊢ Ⅎ 𝑥 ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
68 |
|
nfcv |
⊢ Ⅎ 𝑥 ∅ |
69 |
67 68
|
nfne |
⊢ Ⅎ 𝑥 ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ |
70 |
65 69
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑢 ≠ 𝑧 ∧ ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
71 |
64 70
|
nfrex |
⊢ Ⅎ 𝑥 ∃ 𝑧 ∈ 𝐴 ( 𝑢 ≠ 𝑧 ∧ ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
72 |
|
nfv |
⊢ Ⅎ 𝑢 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
73 |
|
neeq1 |
⊢ ( 𝑢 = 𝑥 → ( 𝑢 ≠ 𝑧 ↔ 𝑥 ≠ 𝑧 ) ) |
74 |
|
csbeq1 |
⊢ ( 𝑢 = 𝑥 → ⦋ 𝑢 / 𝑥 ⦌ 𝐵 = ⦋ 𝑥 / 𝑥 ⦌ 𝐵 ) |
75 |
|
csbid |
⊢ ⦋ 𝑥 / 𝑥 ⦌ 𝐵 = 𝐵 |
76 |
74 75
|
eqtrdi |
⊢ ( 𝑢 = 𝑥 → ⦋ 𝑢 / 𝑥 ⦌ 𝐵 = 𝐵 ) |
77 |
76
|
ineq1d |
⊢ ( 𝑢 = 𝑥 → ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) = ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ) |
78 |
77
|
neeq1d |
⊢ ( 𝑢 = 𝑥 → ( ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ↔ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
79 |
73 78
|
anbi12d |
⊢ ( 𝑢 = 𝑥 → ( ( 𝑢 ≠ 𝑧 ∧ ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ↔ ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
80 |
79
|
rexbidv |
⊢ ( 𝑢 = 𝑥 → ( ∃ 𝑧 ∈ 𝐴 ( 𝑢 ≠ 𝑧 ∧ ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ↔ ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) ) |
81 |
71 72 80
|
cbvrexw |
⊢ ( ∃ 𝑢 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑢 ≠ 𝑧 ∧ ( ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
82 |
63 81
|
bitri |
⊢ ( ¬ Disj 𝑢 ∈ 𝐴 ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
83 |
|
nfcv |
⊢ Ⅎ 𝑢 𝐵 |
84 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑢 → 𝐵 = ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ) |
85 |
83 66 84
|
cbvdisj |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑢 ∈ 𝐴 ⦋ 𝑢 / 𝑥 ⦌ 𝐵 ) |
86 |
82 85
|
xchnxbir |
⊢ ( ¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 ∈ 𝐴 ( 𝑥 ≠ 𝑧 ∧ ( 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) ) |
87 |
61 86
|
sylibr |
⊢ ( ¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ¬ Disj 𝑥 ∈ 𝐴 𝐵 ) |
88 |
87
|
con4i |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran 𝐹 𝑦 ) |