Step |
Hyp |
Ref |
Expression |
1 |
|
disjinfi.b |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) |
2 |
|
disjinfi.d |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝐵 ) |
3 |
|
disjinfi.c |
⊢ ( 𝜑 → 𝐶 ∈ Fin ) |
4 |
|
inss2 |
⊢ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ⊆ 𝐶 |
5 |
|
ssfi |
⊢ ( ( 𝐶 ∈ Fin ∧ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ⊆ 𝐶 ) → ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∈ Fin ) |
6 |
3 4 5
|
sylancl |
⊢ ( 𝜑 → ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∈ Fin ) |
7 |
4
|
a1i |
⊢ ( 𝜑 → ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ⊆ 𝐶 ) |
8 |
3 7
|
ssexd |
⊢ ( 𝜑 → ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∈ V ) |
9 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) → 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
10 |
|
eluni2 |
⊢ ( 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ∈ 𝑤 ) |
11 |
10
|
biimpi |
⊢ ( 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ∃ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ∈ 𝑤 ) |
12 |
|
eqid |
⊢ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
13 |
12
|
elrnmpt |
⊢ ( 𝑤 ∈ V → ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) ) |
14 |
13
|
elv |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) |
15 |
14
|
biimpi |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) |
16 |
15
|
adantr |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ 𝑦 ∈ 𝑤 ) → ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) |
17 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
18 |
17
|
nfrn |
⊢ Ⅎ 𝑥 ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
19 |
18
|
nfcri |
⊢ Ⅎ 𝑥 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
20 |
|
nfv |
⊢ Ⅎ 𝑥 𝑦 ∈ 𝑤 |
21 |
19 20
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ 𝑦 ∈ 𝑤 ) |
22 |
|
simpl |
⊢ ( ( 𝑦 ∈ 𝑤 ∧ 𝑤 = 𝐵 ) → 𝑦 ∈ 𝑤 ) |
23 |
|
simpr |
⊢ ( ( 𝑦 ∈ 𝑤 ∧ 𝑤 = 𝐵 ) → 𝑤 = 𝐵 ) |
24 |
22 23
|
eleqtrd |
⊢ ( ( 𝑦 ∈ 𝑤 ∧ 𝑤 = 𝐵 ) → 𝑦 ∈ 𝐵 ) |
25 |
24
|
ex |
⊢ ( 𝑦 ∈ 𝑤 → ( 𝑤 = 𝐵 → 𝑦 ∈ 𝐵 ) ) |
26 |
25
|
a1d |
⊢ ( 𝑦 ∈ 𝑤 → ( 𝑥 ∈ 𝐴 → ( 𝑤 = 𝐵 → 𝑦 ∈ 𝐵 ) ) ) |
27 |
26
|
adantl |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ 𝑦 ∈ 𝑤 ) → ( 𝑥 ∈ 𝐴 → ( 𝑤 = 𝐵 → 𝑦 ∈ 𝐵 ) ) ) |
28 |
21 27
|
reximdai |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ 𝑦 ∈ 𝑤 ) → ( ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) ) |
29 |
16 28
|
mpd |
⊢ ( ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∧ 𝑦 ∈ 𝑤 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
30 |
29
|
ex |
⊢ ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑦 ∈ 𝑤 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) ) |
31 |
30
|
a1i |
⊢ ( 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( 𝑦 ∈ 𝑤 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) ) ) |
32 |
31
|
rexlimdv |
⊢ ( 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ( ∃ 𝑤 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) 𝑦 ∈ 𝑤 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) ) |
33 |
11 32
|
mpd |
⊢ ( 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
34 |
9 33
|
syl |
⊢ ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
35 |
34
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) |
36 |
|
nfv |
⊢ Ⅎ 𝑥 𝜑 |
37 |
18
|
nfuni |
⊢ Ⅎ 𝑥 ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
38 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐶 |
39 |
37 38
|
nfin |
⊢ Ⅎ 𝑥 ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) |
40 |
39
|
nfcri |
⊢ Ⅎ 𝑥 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) |
41 |
36 40
|
nfan |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) |
42 |
|
nfre1 |
⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) |
43 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) → 𝑦 ∈ 𝐶 ) |
44 |
|
simp2 |
⊢ ( ( 𝑦 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → 𝑥 ∈ 𝐴 ) |
45 |
|
simpr |
⊢ ( ( 𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐵 ) |
46 |
|
simpl |
⊢ ( ( 𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ 𝐶 ) |
47 |
45 46
|
elind |
⊢ ( ( 𝑦 ∈ 𝐶 ∧ 𝑦 ∈ 𝐵 ) → 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
48 |
|
rspe |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
49 |
44 47 48
|
3imp3i2an |
⊢ ( ( 𝑦 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
50 |
49
|
3exp |
⊢ ( 𝑦 ∈ 𝐶 → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
51 |
43 50
|
syl |
⊢ ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
52 |
51
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
53 |
41 42 52
|
rexlimd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
54 |
35 53
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
55 |
|
disjors |
⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
56 |
2 55
|
sylib |
⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
57 |
|
nfv |
⊢ Ⅎ 𝑧 ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
58 |
|
nfcv |
⊢ Ⅎ 𝑥 𝐴 |
59 |
|
nfv |
⊢ Ⅎ 𝑥 𝑧 = 𝑤 |
60 |
|
nfcsb1v |
⊢ Ⅎ 𝑥 ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
61 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑤 |
62 |
61
|
nfcsb1 |
⊢ Ⅎ 𝑥 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 |
63 |
60 62
|
nfin |
⊢ Ⅎ 𝑥 ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
64 |
63
|
nfeq1 |
⊢ Ⅎ 𝑥 ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ |
65 |
59 64
|
nfor |
⊢ Ⅎ 𝑥 ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
66 |
58 65
|
nfralw |
⊢ Ⅎ 𝑥 ∀ 𝑤 ∈ 𝐴 ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
67 |
|
equequ1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 = 𝑤 ↔ 𝑧 = 𝑤 ) ) |
68 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑧 → 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ) |
69 |
68
|
ineq1d |
⊢ ( 𝑥 = 𝑧 → ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) ) |
70 |
69
|
eqeq1d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ↔ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
71 |
67 70
|
orbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ↔ ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) ) |
72 |
71
|
ralbidv |
⊢ ( 𝑥 = 𝑧 → ( ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ↔ ∀ 𝑤 ∈ 𝐴 ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) ) |
73 |
57 66 72
|
cbvralw |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ↔ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑧 = 𝑤 ∨ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
74 |
56 73
|
sylibr |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
75 |
74
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
76 |
|
rspa |
⊢ ( ( ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ∧ 𝑤 ∈ 𝐴 ) → ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ) |
77 |
76
|
orcomd |
⊢ ( ( ∀ 𝑤 ∈ 𝐴 ( 𝑥 = 𝑤 ∨ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ∨ 𝑥 = 𝑤 ) ) |
78 |
75 77
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ∨ 𝑥 = 𝑤 ) ) |
79 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) → 𝑦 ∈ 𝐵 ) |
80 |
|
sbsbc |
⊢ ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
81 |
|
sbcel2 |
⊢ ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) ) |
82 |
|
csbin |
⊢ ⦋ 𝑤 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) = ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐶 ) |
83 |
82
|
eleq2i |
⊢ ( 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐶 ) ) |
84 |
80 81 83
|
3bitri |
⊢ ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐶 ) ) |
85 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐶 ) → 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
86 |
84 85
|
sylbi |
⊢ ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) → 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
87 |
|
inelcm |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) → ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) ≠ ∅ ) |
88 |
87
|
neneqd |
⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) → ¬ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
89 |
79 86 88
|
syl2an |
⊢ ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → ¬ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ) |
90 |
|
pm2.53 |
⊢ ( ( ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ ∨ 𝑥 = 𝑤 ) → ( ¬ ( 𝐵 ∩ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ∅ → 𝑥 = 𝑤 ) ) |
91 |
78 89 90
|
syl2im |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ∧ 𝑤 ∈ 𝐴 ) → ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ) |
92 |
91
|
ralrimiva |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ) |
93 |
92
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ) |
94 |
93
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ) |
95 |
|
reu2 |
⊢ ( ∃! 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ) ) |
96 |
54 94 95
|
sylanbrc |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ∃! 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
97 |
|
riotacl2 |
⊢ ( ∃! 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) → ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) } ) |
98 |
|
nfriota1 |
⊢ Ⅎ 𝑥 ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
99 |
98
|
nfcsb1 |
⊢ Ⅎ 𝑥 ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 |
100 |
99 38
|
nfin |
⊢ Ⅎ 𝑥 ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
101 |
100
|
nfcri |
⊢ Ⅎ 𝑥 𝑦 ∈ ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
102 |
|
csbeq1a |
⊢ ( 𝑥 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝐵 = ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ) |
103 |
102
|
ineq1d |
⊢ ( 𝑥 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → ( 𝐵 ∩ 𝐶 ) = ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
104 |
103
|
eleq2d |
⊢ ( 𝑥 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) |
105 |
98 58 101 104
|
elrabf |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) } ↔ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ 𝐴 ∧ 𝑦 ∈ ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) |
106 |
105
|
simplbi |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) } → ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ 𝐴 ) |
107 |
105
|
simprbi |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) } → 𝑦 ∈ ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
108 |
107
|
ne0d |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) } → ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ) |
109 |
|
nfcv |
⊢ Ⅎ 𝑥 ∅ |
110 |
100 109
|
nfne |
⊢ Ⅎ 𝑥 ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ |
111 |
103
|
neeq1d |
⊢ ( 𝑥 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → ( ( 𝐵 ∩ 𝐶 ) ≠ ∅ ↔ ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
112 |
98 58 110 111
|
elrabf |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ↔ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ 𝐴 ∧ ( ⦋ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
113 |
106 108 112
|
sylanbrc |
⊢ ( ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) } → ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) |
114 |
96 97 113
|
3syl |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) |
115 |
114
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) |
116 |
62 38
|
nfin |
⊢ Ⅎ 𝑥 ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
117 |
116 109
|
nfne |
⊢ Ⅎ 𝑥 ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ |
118 |
|
csbeq1a |
⊢ ( 𝑥 = 𝑤 → 𝐵 = ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
119 |
118
|
ineq1d |
⊢ ( 𝑥 = 𝑤 → ( 𝐵 ∩ 𝐶 ) = ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
120 |
119
|
neeq1d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝐵 ∩ 𝐶 ) ≠ ∅ ↔ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
121 |
61 58 117 120
|
elrabf |
⊢ ( 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ↔ ( 𝑤 ∈ 𝐴 ∧ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ) ) |
122 |
121
|
simprbi |
⊢ ( 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } → ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ) |
123 |
|
n0 |
⊢ ( ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
124 |
122 123
|
sylib |
⊢ ( 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } → ∃ 𝑦 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
125 |
124
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) → ∃ 𝑦 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
126 |
121
|
simplbi |
⊢ ( 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } → 𝑤 ∈ 𝐴 ) |
127 |
|
elinel1 |
⊢ ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) → 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
128 |
127
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
129 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 ∈ 𝐴 ) |
130 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) |
131 |
62
|
nfel1 |
⊢ Ⅎ 𝑥 ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 |
132 |
130 131
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) |
133 |
|
eleq1w |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 ∈ 𝐴 ↔ 𝑤 ∈ 𝐴 ) ) |
134 |
133
|
anbi2d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ) ) |
135 |
118
|
eleq1d |
⊢ ( 𝑥 = 𝑤 → ( 𝐵 ∈ 𝑉 ↔ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) ) |
136 |
134 135
|
imbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 ) ↔ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) ) ) |
137 |
132 136 1
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) |
138 |
137
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) |
139 |
|
eqid |
⊢ ( 𝑤 ∈ 𝐴 ↦ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ( 𝑤 ∈ 𝐴 ↦ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
140 |
139
|
elrnmpt1 |
⊢ ( ( 𝑤 ∈ 𝐴 ∧ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ 𝑉 ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ ran ( 𝑤 ∈ 𝐴 ↦ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) ) |
141 |
129 138 140
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ ran ( 𝑤 ∈ 𝐴 ↦ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) ) |
142 |
|
nfcv |
⊢ Ⅎ 𝑤 𝐵 |
143 |
118
|
equcoms |
⊢ ( 𝑤 = 𝑥 → 𝐵 = ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) |
144 |
143
|
eqcomd |
⊢ ( 𝑤 = 𝑥 → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 = 𝐵 ) |
145 |
62 142 144
|
cbvmpt |
⊢ ( 𝑤 ∈ 𝐴 ↦ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
146 |
145
|
rneqi |
⊢ ran ( 𝑤 ∈ 𝐴 ↦ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ) = ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
147 |
141 146
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
148 |
|
elunii |
⊢ ( ( 𝑦 ∈ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∧ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∈ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) → 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
149 |
128 147 148
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑦 ∈ ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
150 |
|
elinel2 |
⊢ ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) → 𝑦 ∈ 𝐶 ) |
151 |
150
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑦 ∈ 𝐶 ) |
152 |
149 151
|
elind |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) |
153 |
|
nfv |
⊢ Ⅎ 𝑤 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) |
154 |
116
|
nfcri |
⊢ Ⅎ 𝑥 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
155 |
119
|
eleq2d |
⊢ ( 𝑥 = 𝑤 → ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) |
156 |
153 154 155
|
cbvriotaw |
⊢ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) = ( ℩ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
157 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
158 |
|
rspe |
⊢ ( ( 𝑤 ∈ 𝐴 ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ∃ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
159 |
158
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ∃ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
160 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝜑 ) |
161 |
|
sbequ |
⊢ ( 𝑤 = 𝑧 → ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
162 |
|
sbsbc |
⊢ ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) |
163 |
162
|
a1i |
⊢ ( 𝑤 = 𝑧 → ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
164 |
|
sbcel2 |
⊢ ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ⦋ 𝑧 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) ) |
165 |
|
csbin |
⊢ ⦋ 𝑧 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) = ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐶 ) |
166 |
|
csbconstg |
⊢ ( 𝑧 ∈ V → ⦋ 𝑧 / 𝑥 ⦌ 𝐶 = 𝐶 ) |
167 |
166
|
elv |
⊢ ⦋ 𝑧 / 𝑥 ⦌ 𝐶 = 𝐶 |
168 |
167
|
ineq2i |
⊢ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑥 ⦌ 𝐶 ) = ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
169 |
165 168
|
eqtri |
⊢ ⦋ 𝑧 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) = ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
170 |
169
|
eleq2i |
⊢ ( 𝑦 ∈ ⦋ 𝑧 / 𝑥 ⦌ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
171 |
164 170
|
bitri |
⊢ ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
172 |
171
|
a1i |
⊢ ( 𝑤 = 𝑧 → ( [ 𝑧 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) |
173 |
161 163 172
|
3bitrd |
⊢ ( 𝑤 = 𝑧 → ( [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) |
174 |
173
|
anbi2d |
⊢ ( 𝑤 = 𝑧 → ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) ) |
175 |
|
equequ2 |
⊢ ( 𝑤 = 𝑧 → ( 𝑥 = 𝑤 ↔ 𝑥 = 𝑧 ) ) |
176 |
174 175
|
imbi12d |
⊢ ( 𝑤 = 𝑧 → ( ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) ) ) |
177 |
176
|
cbvralvw |
⊢ ( ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) ) |
178 |
177
|
ralbii |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) ) |
179 |
|
nfv |
⊢ Ⅎ 𝑤 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) |
180 |
60 38
|
nfin |
⊢ Ⅎ 𝑥 ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
181 |
180
|
nfcri |
⊢ Ⅎ 𝑥 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
182 |
154 181
|
nfan |
⊢ Ⅎ 𝑥 ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
183 |
|
nfv |
⊢ Ⅎ 𝑥 𝑤 = 𝑧 |
184 |
182 183
|
nfim |
⊢ Ⅎ 𝑥 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) |
185 |
58 184
|
nfralw |
⊢ Ⅎ 𝑥 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) |
186 |
155
|
anbi1d |
⊢ ( 𝑥 = 𝑤 → ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) ) |
187 |
|
equequ1 |
⊢ ( 𝑥 = 𝑤 → ( 𝑥 = 𝑧 ↔ 𝑤 = 𝑧 ) ) |
188 |
186 187
|
imbi12d |
⊢ ( 𝑥 = 𝑤 → ( ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) ↔ ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) ) |
189 |
188
|
ralbidv |
⊢ ( 𝑥 = 𝑤 → ( ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) ) |
190 |
179 185 189
|
cbvralw |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑧 ) ↔ ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) |
191 |
|
sbsbc |
⊢ ( [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ↔ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
192 |
|
sbcel2 |
⊢ ( [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ⦋ 𝑧 / 𝑤 ⦌ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
193 |
|
csbin |
⊢ ⦋ 𝑧 / 𝑤 ⦌ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) = ( ⦋ 𝑧 / 𝑤 ⦌ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑤 ⦌ 𝐶 ) |
194 |
|
csbcow |
⊢ ⦋ 𝑧 / 𝑤 ⦌ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 = ⦋ 𝑧 / 𝑥 ⦌ 𝐵 |
195 |
|
csbconstg |
⊢ ( 𝑧 ∈ V → ⦋ 𝑧 / 𝑤 ⦌ 𝐶 = 𝐶 ) |
196 |
195
|
elv |
⊢ ⦋ 𝑧 / 𝑤 ⦌ 𝐶 = 𝐶 |
197 |
194 196
|
ineq12i |
⊢ ( ⦋ 𝑧 / 𝑤 ⦌ ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ ⦋ 𝑧 / 𝑤 ⦌ 𝐶 ) = ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
198 |
193 197
|
eqtri |
⊢ ⦋ 𝑧 / 𝑤 ⦌ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) = ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) |
199 |
198
|
eleq2i |
⊢ ( 𝑦 ∈ ⦋ 𝑧 / 𝑤 ⦌ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ↔ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
200 |
191 192 199
|
3bitrri |
⊢ ( 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ↔ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
201 |
200
|
anbi2i |
⊢ ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ) |
202 |
201
|
imbi1i |
⊢ ( ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ↔ ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) |
203 |
202
|
2ralbii |
⊢ ( ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ 𝑦 ∈ ( ⦋ 𝑧 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ↔ ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) |
204 |
178 190 203
|
3bitri |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( ( 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ∧ [ 𝑤 / 𝑥 ] 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) → 𝑥 = 𝑤 ) ↔ ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) |
205 |
94 204
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) |
206 |
160 152 205
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) |
207 |
|
reu2 |
⊢ ( ∃! 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ↔ ( ∃ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ ∀ 𝑤 ∈ 𝐴 ∀ 𝑧 ∈ 𝐴 ( ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ∧ [ 𝑧 / 𝑤 ] 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = 𝑧 ) ) ) |
208 |
159 206 207
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ∃! 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) |
209 |
|
riota1 |
⊢ ( ∃! 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) → ( ( 𝑤 ∈ 𝐴 ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ↔ ( ℩ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) = 𝑤 ) ) |
210 |
208 209
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ( ( 𝑤 ∈ 𝐴 ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) ↔ ( ℩ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) = 𝑤 ) ) |
211 |
129 157 210
|
mpbi2and |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ( ℩ 𝑤 ∈ 𝐴 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) = 𝑤 ) |
212 |
156 211
|
eqtr2id |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
213 |
152 212
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) ∧ 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) ) → ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
214 |
213
|
ex |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ 𝐴 ) → ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) → ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) ) |
215 |
126 214
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) → ( 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) → ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) ) |
216 |
215
|
eximdv |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) → ( ∃ 𝑦 𝑦 ∈ ( ⦋ 𝑤 / 𝑥 ⦌ 𝐵 ∩ 𝐶 ) → ∃ 𝑦 ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) ) |
217 |
125 216
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) → ∃ 𝑦 ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
218 |
|
df-rex |
⊢ ( ∃ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ ∃ 𝑦 ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∧ 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
219 |
217 218
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) → ∃ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
220 |
219
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ∃ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
221 |
|
eqid |
⊢ ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ↦ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) = ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ↦ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) |
222 |
221
|
fompt |
⊢ ( ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ↦ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) : ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) –onto→ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ↔ ( ∀ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ∧ ∀ 𝑤 ∈ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ∃ 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) 𝑤 = ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) ) |
223 |
115 220 222
|
sylanbrc |
⊢ ( 𝜑 → ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ↦ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) : ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) –onto→ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ) |
224 |
|
fodomg |
⊢ ( ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∈ V → ( ( 𝑦 ∈ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ↦ ( ℩ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ∩ 𝐶 ) ) ) : ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) –onto→ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } → { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ≼ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) ) |
225 |
8 223 224
|
sylc |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ≼ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) |
226 |
|
domfi |
⊢ ( ( ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ∈ Fin ∧ { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ≼ ( ∪ ran ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∩ 𝐶 ) ) → { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ∈ Fin ) |
227 |
6 225 226
|
syl2anc |
⊢ ( 𝜑 → { 𝑥 ∈ 𝐴 ∣ ( 𝐵 ∩ 𝐶 ) ≠ ∅ } ∈ Fin ) |