Step |
Hyp |
Ref |
Expression |
1 |
|
kqval.2 |
⊢ 𝐹 = ( 𝑥 ∈ 𝑋 ↦ { 𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦 } ) |
2 |
|
regr1lem.2 |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
3 |
|
regr1lem.3 |
⊢ ( 𝜑 → 𝐽 ∈ Reg ) |
4 |
|
regr1lem.4 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑋 ) |
5 |
|
regr1lem.5 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑋 ) |
6 |
|
regr1lem.6 |
⊢ ( 𝜑 → 𝑈 ∈ 𝐽 ) |
7 |
|
regr1lem.7 |
⊢ ( 𝜑 → ¬ ∃ 𝑚 ∈ ( KQ ‘ 𝐽 ) ∃ 𝑛 ∈ ( KQ ‘ 𝐽 ) ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( 𝑚 ∩ 𝑛 ) = ∅ ) ) |
8 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝐽 ∈ Reg ) |
9 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝑈 ∈ 𝐽 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝐴 ∈ 𝑈 ) |
11 |
|
regsep |
⊢ ( ( 𝐽 ∈ Reg ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ) → ∃ 𝑧 ∈ 𝐽 ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) |
12 |
8 9 10 11
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → ∃ 𝑧 ∈ 𝐽 ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) |
13 |
7
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) → ¬ ∃ 𝑚 ∈ ( KQ ‘ 𝐽 ) ∃ 𝑛 ∈ ( KQ ‘ 𝐽 ) ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( 𝑚 ∩ 𝑛 ) = ∅ ) ) |
14 |
2
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
15 |
|
simplrl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝑧 ∈ 𝐽 ) |
16 |
1
|
kqopn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑧 ∈ 𝐽 ) → ( 𝐹 “ 𝑧 ) ∈ ( KQ ‘ 𝐽 ) ) |
17 |
14 15 16
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝐹 “ 𝑧 ) ∈ ( KQ ‘ 𝐽 ) ) |
18 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
19 |
14 18
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝑋 = ∪ 𝐽 ) |
20 |
19
|
difeq1d |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) = ( ∪ 𝐽 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) |
21 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
22 |
14 21
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝐽 ∈ Top ) |
23 |
|
elssuni |
⊢ ( 𝑧 ∈ 𝐽 → 𝑧 ⊆ ∪ 𝐽 ) |
24 |
15 23
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝑧 ⊆ ∪ 𝐽 ) |
25 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
26 |
25
|
clscld |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑧 ⊆ ∪ 𝐽 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ∈ ( Clsd ‘ 𝐽 ) ) |
27 |
22 24 26
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ∈ ( Clsd ‘ 𝐽 ) ) |
28 |
25
|
cldopn |
⊢ ( ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ∈ ( Clsd ‘ 𝐽 ) → ( ∪ 𝐽 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ∈ 𝐽 ) |
29 |
27 28
|
syl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( ∪ 𝐽 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ∈ 𝐽 ) |
30 |
20 29
|
eqeltrd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ∈ 𝐽 ) |
31 |
1
|
kqopn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ∈ 𝐽 ) → ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ∈ ( KQ ‘ 𝐽 ) ) |
32 |
14 30 31
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ∈ ( KQ ‘ 𝐽 ) ) |
33 |
|
simprrl |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) → 𝐴 ∈ 𝑧 ) |
34 |
33
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝐴 ∈ 𝑧 ) |
35 |
4
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝐴 ∈ 𝑋 ) |
36 |
1
|
kqfvima |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑧 ∈ 𝐽 ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 ∈ 𝑧 ↔ ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ) ) |
37 |
14 15 35 36
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝐴 ∈ 𝑧 ↔ ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ) ) |
38 |
34 37
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ) |
39 |
5
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝐵 ∈ 𝑋 ) |
40 |
|
simprrr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) → ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) |
41 |
40
|
sseld |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) → ( 𝐵 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) → 𝐵 ∈ 𝑈 ) ) |
42 |
41
|
con3dimp |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ¬ 𝐵 ∈ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) |
43 |
39 42
|
eldifd |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝐵 ∈ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) |
44 |
1
|
kqfvima |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ∈ 𝐽 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐵 ∈ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ↔ ( 𝐹 ‘ 𝐵 ) ∈ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ) |
45 |
14 30 39 44
|
syl3anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝐵 ∈ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ↔ ( 𝐹 ‘ 𝐵 ) ∈ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ) |
46 |
43 45
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝐹 ‘ 𝐵 ) ∈ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) |
47 |
25
|
sscls |
⊢ ( ( 𝐽 ∈ Top ∧ 𝑧 ⊆ ∪ 𝐽 ) → 𝑧 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) |
48 |
22 24 47
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → 𝑧 ⊆ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) |
49 |
48
|
sscond |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ⊆ ( 𝑋 ∖ 𝑧 ) ) |
50 |
|
imass2 |
⊢ ( ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ⊆ ( 𝑋 ∖ 𝑧 ) → ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ⊆ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) |
51 |
|
sslin |
⊢ ( ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ⊆ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) → ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ⊆ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) ) |
52 |
49 50 51
|
3syl |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ⊆ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) ) |
53 |
1
|
kqdisj |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝑧 ∈ 𝐽 ) → ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) = ∅ ) |
54 |
14 15 53
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) = ∅ ) |
55 |
|
sseq0 |
⊢ ( ( ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ⊆ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) ∧ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ 𝑧 ) ) ) = ∅ ) → ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) = ∅ ) |
56 |
52 54 55
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) = ∅ ) |
57 |
|
eleq2 |
⊢ ( 𝑚 = ( 𝐹 “ 𝑧 ) → ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ↔ ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ) ) |
58 |
|
ineq1 |
⊢ ( 𝑚 = ( 𝐹 “ 𝑧 ) → ( 𝑚 ∩ 𝑛 ) = ( ( 𝐹 “ 𝑧 ) ∩ 𝑛 ) ) |
59 |
58
|
eqeq1d |
⊢ ( 𝑚 = ( 𝐹 “ 𝑧 ) → ( ( 𝑚 ∩ 𝑛 ) = ∅ ↔ ( ( 𝐹 “ 𝑧 ) ∩ 𝑛 ) = ∅ ) ) |
60 |
57 59
|
3anbi13d |
⊢ ( 𝑚 = ( 𝐹 “ 𝑧 ) → ( ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( 𝑚 ∩ 𝑛 ) = ∅ ) ↔ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( ( 𝐹 “ 𝑧 ) ∩ 𝑛 ) = ∅ ) ) ) |
61 |
|
eleq2 |
⊢ ( 𝑛 = ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) → ( ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ↔ ( 𝐹 ‘ 𝐵 ) ∈ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ) |
62 |
|
ineq2 |
⊢ ( 𝑛 = ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) → ( ( 𝐹 “ 𝑧 ) ∩ 𝑛 ) = ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) ) |
63 |
62
|
eqeq1d |
⊢ ( 𝑛 = ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) → ( ( ( 𝐹 “ 𝑧 ) ∩ 𝑛 ) = ∅ ↔ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) = ∅ ) ) |
64 |
61 63
|
3anbi23d |
⊢ ( 𝑛 = ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) → ( ( ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( ( 𝐹 “ 𝑧 ) ∩ 𝑛 ) = ∅ ) ↔ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ∧ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) = ∅ ) ) ) |
65 |
60 64
|
rspc2ev |
⊢ ( ( ( 𝐹 “ 𝑧 ) ∈ ( KQ ‘ 𝐽 ) ∧ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ∈ ( KQ ‘ 𝐽 ) ∧ ( ( 𝐹 ‘ 𝐴 ) ∈ ( 𝐹 “ 𝑧 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ∧ ( ( 𝐹 “ 𝑧 ) ∩ ( 𝐹 “ ( 𝑋 ∖ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ) ) ) = ∅ ) ) → ∃ 𝑚 ∈ ( KQ ‘ 𝐽 ) ∃ 𝑛 ∈ ( KQ ‘ 𝐽 ) ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( 𝑚 ∩ 𝑛 ) = ∅ ) ) |
66 |
17 32 38 46 56 65
|
syl113anc |
⊢ ( ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) ∧ ¬ 𝐵 ∈ 𝑈 ) → ∃ 𝑚 ∈ ( KQ ‘ 𝐽 ) ∃ 𝑛 ∈ ( KQ ‘ 𝐽 ) ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( 𝑚 ∩ 𝑛 ) = ∅ ) ) |
67 |
66
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) → ( ¬ 𝐵 ∈ 𝑈 → ∃ 𝑚 ∈ ( KQ ‘ 𝐽 ) ∃ 𝑛 ∈ ( KQ ‘ 𝐽 ) ( ( 𝐹 ‘ 𝐴 ) ∈ 𝑚 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑛 ∧ ( 𝑚 ∩ 𝑛 ) = ∅ ) ) ) |
68 |
13 67
|
mt3d |
⊢ ( ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) ∧ ( 𝑧 ∈ 𝐽 ∧ ( 𝐴 ∈ 𝑧 ∧ ( ( cls ‘ 𝐽 ) ‘ 𝑧 ) ⊆ 𝑈 ) ) ) → 𝐵 ∈ 𝑈 ) |
69 |
12 68
|
rexlimddv |
⊢ ( ( 𝜑 ∧ 𝐴 ∈ 𝑈 ) → 𝐵 ∈ 𝑈 ) |
70 |
69
|
ex |
⊢ ( 𝜑 → ( 𝐴 ∈ 𝑈 → 𝐵 ∈ 𝑈 ) ) |