Step |
Hyp |
Ref |
Expression |
1 |
|
ptcls.2 |
⊢ 𝐽 = ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ) |
2 |
|
ptcls.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
ptcls.j |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑅 ∈ ( TopOn ‘ 𝑋 ) ) |
4 |
|
ptcls.c |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑆 ⊆ 𝑋 ) |
5 |
|
ptclsg.1 |
⊢ ( 𝜑 → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴 ) |
6 |
|
topontop |
⊢ ( 𝑅 ∈ ( TopOn ‘ 𝑋 ) → 𝑅 ∈ Top ) |
7 |
3 6
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑅 ∈ Top ) |
8 |
|
toponuni |
⊢ ( 𝑅 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝑅 ) |
9 |
3 8
|
syl |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑋 = ∪ 𝑅 ) |
10 |
4 9
|
sseqtrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑆 ⊆ ∪ 𝑅 ) |
11 |
|
eqid |
⊢ ∪ 𝑅 = ∪ 𝑅 |
12 |
11
|
clscld |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ) → ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝑅 ) ) |
13 |
7 10 12
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝑅 ) ) |
14 |
2 7 13
|
ptcldmpt |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ∈ ( Clsd ‘ ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ) ) ) |
15 |
1
|
fveq2i |
⊢ ( Clsd ‘ 𝐽 ) = ( Clsd ‘ ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ) ) |
16 |
14 15
|
eleqtrrdi |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ) |
17 |
11
|
sscls |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ) → 𝑆 ⊆ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
18 |
7 10 17
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑆 ⊆ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
19 |
18
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝑆 ⊆ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
20 |
|
ss2ixp |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝑆 ⊆ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) → X 𝑘 ∈ 𝐴 𝑆 ⊆ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝑆 ⊆ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
22 |
|
eqid |
⊢ ∪ 𝐽 = ∪ 𝐽 |
23 |
22
|
clsss2 |
⊢ ( ( X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ∈ ( Clsd ‘ 𝐽 ) ∧ X 𝑘 ∈ 𝐴 𝑆 ⊆ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( ( cls ‘ 𝐽 ) ‘ X 𝑘 ∈ 𝐴 𝑆 ) ⊆ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
24 |
16 21 23
|
syl2anc |
⊢ ( 𝜑 → ( ( cls ‘ 𝐽 ) ‘ X 𝑘 ∈ 𝐴 𝑆 ) ⊆ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
25 |
|
vex |
⊢ 𝑢 ∈ V |
26 |
|
eqeq1 |
⊢ ( 𝑥 = 𝑢 → ( 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ↔ 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) |
27 |
26
|
anbi2d |
⊢ ( 𝑥 = 𝑢 → ( ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ↔ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) ) |
28 |
27
|
exbidv |
⊢ ( 𝑥 = 𝑢 → ( ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ↔ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) ) |
29 |
25 28
|
elab |
⊢ ( 𝑢 ∈ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ↔ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) |
30 |
|
nffvmpt1 |
⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) |
31 |
30
|
nfel2 |
⊢ Ⅎ 𝑘 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) |
32 |
|
nfv |
⊢ Ⅎ 𝑦 ( 𝑔 ‘ 𝑘 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) |
33 |
|
fveq2 |
⊢ ( 𝑦 = 𝑘 → ( 𝑔 ‘ 𝑦 ) = ( 𝑔 ‘ 𝑘 ) ) |
34 |
|
fveq2 |
⊢ ( 𝑦 = 𝑘 → ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) ) |
35 |
33 34
|
eleq12d |
⊢ ( 𝑦 = 𝑘 → ( ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ↔ ( 𝑔 ‘ 𝑘 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) ) ) |
36 |
31 32 35
|
cbvralw |
⊢ ( ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) ) |
37 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ 𝐴 ) |
38 |
|
eqid |
⊢ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) = ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) |
39 |
38
|
fvmpt2 |
⊢ ( ( 𝑘 ∈ 𝐴 ∧ 𝑅 ∈ ( TopOn ‘ 𝑋 ) ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) = 𝑅 ) |
40 |
37 3 39
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) = 𝑅 ) |
41 |
40
|
eleq2d |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑔 ‘ 𝑘 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) ↔ ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) |
42 |
41
|
ralbidva |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑘 ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) |
43 |
36 42
|
syl5bb |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ↔ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) |
44 |
43
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ↔ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) ) |
45 |
44
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ↔ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) ) |
46 |
45
|
biimpa |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ) → ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) |
47 |
5
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴 ) |
48 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → 𝜑 ) |
49 |
|
vex |
⊢ 𝑓 ∈ V |
50 |
49
|
elixp |
⊢ ( 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ↔ ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ) |
51 |
50
|
simprbi |
⊢ ( 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) → ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
52 |
51
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |
53 |
11
|
clsndisj |
⊢ ( ( ( 𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) |
54 |
53
|
ex |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) |
55 |
54
|
3expia |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ) → ( ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) → ( ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) ) |
56 |
7 10 55
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) → ( ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) ) |
57 |
56
|
ralimdva |
⊢ ( 𝜑 → ( ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) → ∀ 𝑘 ∈ 𝐴 ( ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) ) |
58 |
48 52 57
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) |
59 |
|
simprlr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) |
60 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) |
61 |
33
|
cbvixpv |
⊢ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) = X 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) |
62 |
60 61
|
eleqtrdi |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → 𝑓 ∈ X 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ) |
63 |
49
|
elixp |
⊢ ( 𝑓 ∈ X 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ↔ ( 𝑓 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) ) |
64 |
63
|
simprbi |
⊢ ( 𝑓 ∈ X 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) → ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) |
65 |
62 64
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) |
66 |
|
r19.26 |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) ↔ ( ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) ) |
67 |
59 65 66
|
sylanbrc |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) ) |
68 |
|
ralim |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) → ( ∀ 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ∧ ( 𝑓 ‘ 𝑘 ) ∈ ( 𝑔 ‘ 𝑘 ) ) → ∀ 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) |
69 |
58 67 68
|
sylc |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) |
70 |
|
rabn0 |
⊢ ( { 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) } ≠ ∅ ↔ ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) |
71 |
|
dfin5 |
⊢ ( ∪ 𝑘 ∈ 𝐴 𝑆 ∩ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) = { 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) } |
72 |
|
inss2 |
⊢ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ⊆ 𝑆 |
73 |
|
ssiun2 |
⊢ ( 𝑘 ∈ 𝐴 → 𝑆 ⊆ ∪ 𝑘 ∈ 𝐴 𝑆 ) |
74 |
72 73
|
sstrid |
⊢ ( 𝑘 ∈ 𝐴 → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ⊆ ∪ 𝑘 ∈ 𝐴 𝑆 ) |
75 |
|
sseqin2 |
⊢ ( ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ⊆ ∪ 𝑘 ∈ 𝐴 𝑆 ↔ ( ∪ 𝑘 ∈ 𝐴 𝑆 ∩ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) = ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) |
76 |
74 75
|
sylib |
⊢ ( 𝑘 ∈ 𝐴 → ( ∪ 𝑘 ∈ 𝐴 𝑆 ∩ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) = ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) |
77 |
71 76
|
eqtr3id |
⊢ ( 𝑘 ∈ 𝐴 → { 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) } = ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) |
78 |
77
|
neeq1d |
⊢ ( 𝑘 ∈ 𝐴 → ( { 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 ∣ 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) } ≠ ∅ ↔ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) |
79 |
70 78
|
bitr3id |
⊢ ( 𝑘 ∈ 𝐴 → ( ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) ) |
80 |
79
|
ralbiia |
⊢ ( ∀ 𝑘 ∈ 𝐴 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ∀ 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) |
81 |
69 80
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑘 ∈ 𝐴 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) |
82 |
|
nfv |
⊢ Ⅎ 𝑦 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) |
83 |
|
nfiu1 |
⊢ Ⅎ 𝑘 ∪ 𝑘 ∈ 𝐴 𝑆 |
84 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑔 ‘ 𝑦 ) |
85 |
|
nfcsb1v |
⊢ Ⅎ 𝑘 ⦋ 𝑦 / 𝑘 ⦌ 𝑆 |
86 |
84 85
|
nfin |
⊢ Ⅎ 𝑘 ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) |
87 |
86
|
nfel2 |
⊢ Ⅎ 𝑘 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) |
88 |
83 87
|
nfrex |
⊢ Ⅎ 𝑘 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) |
89 |
|
fveq2 |
⊢ ( 𝑘 = 𝑦 → ( 𝑔 ‘ 𝑘 ) = ( 𝑔 ‘ 𝑦 ) ) |
90 |
|
csbeq1a |
⊢ ( 𝑘 = 𝑦 → 𝑆 = ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) |
91 |
89 90
|
ineq12d |
⊢ ( 𝑘 = 𝑦 → ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) = ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) |
92 |
91
|
eleq2d |
⊢ ( 𝑘 = 𝑦 → ( 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) ) |
93 |
92
|
rexbidv |
⊢ ( 𝑘 = 𝑦 → ( ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) ) |
94 |
82 88 93
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ 𝐴 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ∀ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) |
95 |
81 94
|
sylib |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∀ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) |
96 |
|
eleq1 |
⊢ ( 𝑧 = ( ℎ ‘ 𝑦 ) → ( 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ↔ ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) ) |
97 |
96
|
acni3 |
⊢ ( ( ∪ 𝑘 ∈ 𝐴 𝑆 ∈ AC 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ∃ 𝑧 ∈ ∪ 𝑘 ∈ 𝐴 𝑆 𝑧 ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) → ∃ ℎ ( ℎ : 𝐴 ⟶ ∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) ) |
98 |
47 95 97
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ∃ ℎ ( ℎ : 𝐴 ⟶ ∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) ) |
99 |
|
ffn |
⊢ ( ℎ : 𝐴 ⟶ ∪ 𝑘 ∈ 𝐴 𝑆 → ℎ Fn 𝐴 ) |
100 |
|
nfv |
⊢ Ⅎ 𝑦 ( ℎ ‘ 𝑘 ) ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) |
101 |
86
|
nfel2 |
⊢ Ⅎ 𝑘 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) |
102 |
|
fveq2 |
⊢ ( 𝑘 = 𝑦 → ( ℎ ‘ 𝑘 ) = ( ℎ ‘ 𝑦 ) ) |
103 |
102 91
|
eleq12d |
⊢ ( 𝑘 = 𝑦 → ( ( ℎ ‘ 𝑘 ) ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) ) |
104 |
100 101 103
|
cbvralw |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( ℎ ‘ 𝑘 ) ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) |
105 |
|
ne0i |
⊢ ( ℎ ∈ X 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) → X 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ) |
106 |
|
vex |
⊢ ℎ ∈ V |
107 |
106
|
elixp |
⊢ ( ℎ ∈ X 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ↔ ( ℎ Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( ℎ ‘ 𝑘 ) ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) ) |
108 |
|
ixpin |
⊢ X 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) = ( X 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) |
109 |
61
|
ineq1i |
⊢ ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) = ( X 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) |
110 |
108 109
|
eqtr4i |
⊢ X 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) = ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) |
111 |
110
|
neeq1i |
⊢ ( X 𝑘 ∈ 𝐴 ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ≠ ∅ ↔ ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) |
112 |
105 107 111
|
3imtr3i |
⊢ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( ℎ ‘ 𝑘 ) ∈ ( ( 𝑔 ‘ 𝑘 ) ∩ 𝑆 ) ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) |
113 |
104 112
|
sylan2br |
⊢ ( ( ℎ Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) |
114 |
99 113
|
sylan |
⊢ ( ( ℎ : 𝐴 ⟶ ∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) |
115 |
114
|
exlimiv |
⊢ ( ∃ ℎ ( ℎ : 𝐴 ⟶ ∪ 𝑘 ∈ 𝐴 𝑆 ∧ ∀ 𝑦 ∈ 𝐴 ( ℎ ‘ 𝑦 ) ∈ ( ( 𝑔 ‘ 𝑦 ) ∩ ⦋ 𝑦 / 𝑘 ⦌ 𝑆 ) ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) |
116 |
98 115
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ∧ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) |
117 |
116
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑘 ∈ 𝐴 ( 𝑔 ‘ 𝑘 ) ∈ 𝑅 ) ) → ( 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) |
118 |
46 117
|
syldan |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ) → ( 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) |
119 |
118
|
3adantr3 |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ) → ( 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) |
120 |
|
eleq2 |
⊢ ( 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( 𝑓 ∈ 𝑢 ↔ 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) ) |
121 |
|
ineq1 |
⊢ ( 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) = ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ) |
122 |
121
|
neeq1d |
⊢ ( 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ↔ ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) |
123 |
120 122
|
imbi12d |
⊢ ( 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ↔ ( 𝑓 ∈ X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) ) |
124 |
119 123
|
syl5ibrcom |
⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) ∧ ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ) → ( 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) → ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) ) |
125 |
124
|
expimpd |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) → ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) ) |
126 |
125
|
exlimdv |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑢 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) → ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) ) |
127 |
29 126
|
syl5bi |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( 𝑢 ∈ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } → ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) ) |
128 |
127
|
ralrimiv |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ∀ 𝑢 ∈ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) |
129 |
7
|
fmpttd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) : 𝐴 ⟶ Top ) |
130 |
129
|
ffnd |
⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) Fn 𝐴 ) |
131 |
|
eqid |
⊢ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } = { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } |
132 |
131
|
ptval |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) Fn 𝐴 ) → ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ) = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
133 |
2 130 132
|
syl2anc |
⊢ ( 𝜑 → ( ∏t ‘ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ) = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
134 |
1 133
|
syl5eq |
⊢ ( 𝜑 → 𝐽 = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
135 |
134
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → 𝐽 = ( topGen ‘ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ) ) |
136 |
3
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝑅 ∈ ( TopOn ‘ 𝑋 ) ) |
137 |
1
|
pttopon |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ∀ 𝑘 ∈ 𝐴 𝑅 ∈ ( TopOn ‘ 𝑋 ) ) → 𝐽 ∈ ( TopOn ‘ X 𝑘 ∈ 𝐴 𝑋 ) ) |
138 |
2 136 137
|
syl2anc |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ X 𝑘 ∈ 𝐴 𝑋 ) ) |
139 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ X 𝑘 ∈ 𝐴 𝑋 ) → X 𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽 ) |
140 |
138 139
|
syl |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽 ) |
141 |
140
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → X 𝑘 ∈ 𝐴 𝑋 = ∪ 𝐽 ) |
142 |
131
|
ptbas |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) : 𝐴 ⟶ Top ) → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases ) |
143 |
2 129 142
|
syl2anc |
⊢ ( 𝜑 → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases ) |
144 |
143
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ∈ TopBases ) |
145 |
4
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 𝑆 ⊆ 𝑋 ) |
146 |
|
ss2ixp |
⊢ ( ∀ 𝑘 ∈ 𝐴 𝑆 ⊆ 𝑋 → X 𝑘 ∈ 𝐴 𝑆 ⊆ X 𝑘 ∈ 𝐴 𝑋 ) |
147 |
145 146
|
syl |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 𝑆 ⊆ X 𝑘 ∈ 𝐴 𝑋 ) |
148 |
147
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → X 𝑘 ∈ 𝐴 𝑆 ⊆ X 𝑘 ∈ 𝐴 𝑋 ) |
149 |
11
|
clsss3 |
⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ⊆ ∪ 𝑅 ) → ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ ∪ 𝑅 ) |
150 |
7 10 149
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ ∪ 𝑅 ) |
151 |
150 9
|
sseqtrrd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ 𝑋 ) |
152 |
151
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ 𝑋 ) |
153 |
|
ss2ixp |
⊢ ( ∀ 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ 𝑋 → X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ X 𝑘 ∈ 𝐴 𝑋 ) |
154 |
152 153
|
syl |
⊢ ( 𝜑 → X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ⊆ X 𝑘 ∈ 𝐴 𝑋 ) |
155 |
154
|
sselda |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → 𝑓 ∈ X 𝑘 ∈ 𝐴 𝑋 ) |
156 |
135 141 144 148 155
|
elcls3 |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → ( 𝑓 ∈ ( ( cls ‘ 𝐽 ) ‘ X 𝑘 ∈ 𝐴 𝑆 ) ↔ ∀ 𝑢 ∈ { 𝑥 ∣ ∃ 𝑔 ( ( 𝑔 Fn 𝐴 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ∈ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ∧ ∃ 𝑧 ∈ Fin ∀ 𝑦 ∈ ( 𝐴 ∖ 𝑧 ) ( 𝑔 ‘ 𝑦 ) = ∪ ( ( 𝑘 ∈ 𝐴 ↦ 𝑅 ) ‘ 𝑦 ) ) ∧ 𝑥 = X 𝑦 ∈ 𝐴 ( 𝑔 ‘ 𝑦 ) ) } ( 𝑓 ∈ 𝑢 → ( 𝑢 ∩ X 𝑘 ∈ 𝐴 𝑆 ) ≠ ∅ ) ) ) |
157 |
128 156
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑓 ∈ X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) → 𝑓 ∈ ( ( cls ‘ 𝐽 ) ‘ X 𝑘 ∈ 𝐴 𝑆 ) ) |
158 |
24 157
|
eqelssd |
⊢ ( 𝜑 → ( ( cls ‘ 𝐽 ) ‘ X 𝑘 ∈ 𝐴 𝑆 ) = X 𝑘 ∈ 𝐴 ( ( cls ‘ 𝑅 ) ‘ 𝑆 ) ) |