| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( ACS ‘ 𝑥 ) = ( ACS ‘ 𝑋 ) ) |
| 2 |
|
pweq |
⊢ ( 𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋 ) |
| 3 |
2
|
fveq2d |
⊢ ( 𝑥 = 𝑋 → ( Moore ‘ 𝒫 𝑥 ) = ( Moore ‘ 𝒫 𝑋 ) ) |
| 4 |
1 3
|
eleq12d |
⊢ ( 𝑥 = 𝑋 → ( ( ACS ‘ 𝑥 ) ∈ ( Moore ‘ 𝒫 𝑥 ) ↔ ( ACS ‘ 𝑋 ) ∈ ( Moore ‘ 𝒫 𝑋 ) ) ) |
| 5 |
|
acsmre |
⊢ ( 𝑎 ∈ ( ACS ‘ 𝑥 ) → 𝑎 ∈ ( Moore ‘ 𝑥 ) ) |
| 6 |
|
mresspw |
⊢ ( 𝑎 ∈ ( Moore ‘ 𝑥 ) → 𝑎 ⊆ 𝒫 𝑥 ) |
| 7 |
5 6
|
syl |
⊢ ( 𝑎 ∈ ( ACS ‘ 𝑥 ) → 𝑎 ⊆ 𝒫 𝑥 ) |
| 8 |
5 7
|
elpwd |
⊢ ( 𝑎 ∈ ( ACS ‘ 𝑥 ) → 𝑎 ∈ 𝒫 𝒫 𝑥 ) |
| 9 |
8
|
ssriv |
⊢ ( ACS ‘ 𝑥 ) ⊆ 𝒫 𝒫 𝑥 |
| 10 |
9
|
a1i |
⊢ ( ⊤ → ( ACS ‘ 𝑥 ) ⊆ 𝒫 𝒫 𝑥 ) |
| 11 |
|
vex |
⊢ 𝑥 ∈ V |
| 12 |
|
mremre |
⊢ ( 𝑥 ∈ V → ( Moore ‘ 𝑥 ) ∈ ( Moore ‘ 𝒫 𝑥 ) ) |
| 13 |
11 12
|
mp1i |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( Moore ‘ 𝑥 ) ∈ ( Moore ‘ 𝒫 𝑥 ) ) |
| 14 |
5
|
ssriv |
⊢ ( ACS ‘ 𝑥 ) ⊆ ( Moore ‘ 𝑥 ) |
| 15 |
|
sstr |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ ( ACS ‘ 𝑥 ) ⊆ ( Moore ‘ 𝑥 ) ) → 𝑎 ⊆ ( Moore ‘ 𝑥 ) ) |
| 16 |
14 15
|
mpan2 |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → 𝑎 ⊆ ( Moore ‘ 𝑥 ) ) |
| 17 |
|
mrerintcl |
⊢ ( ( ( Moore ‘ 𝑥 ) ∈ ( Moore ‘ 𝒫 𝑥 ) ∧ 𝑎 ⊆ ( Moore ‘ 𝑥 ) ) → ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ∈ ( Moore ‘ 𝑥 ) ) |
| 18 |
13 16 17
|
syl2anc |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ∈ ( Moore ‘ 𝑥 ) ) |
| 19 |
|
ssel2 |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → 𝑑 ∈ ( ACS ‘ 𝑥 ) ) |
| 20 |
19
|
acsmred |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → 𝑑 ∈ ( Moore ‘ 𝑥 ) ) |
| 21 |
|
eqid |
⊢ ( mrCls ‘ 𝑑 ) = ( mrCls ‘ 𝑑 ) |
| 22 |
20 21
|
mrcssvd |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ) |
| 23 |
22
|
ralrimiva |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ) |
| 24 |
23
|
adantr |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑐 ∈ 𝒫 𝑥 ) → ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ) |
| 25 |
|
iunss |
⊢ ( ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ↔ ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ) |
| 26 |
24 25
|
sylibr |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑐 ∈ 𝒫 𝑥 ) → ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ) |
| 27 |
11
|
elpw2 |
⊢ ( ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ∈ 𝒫 𝑥 ↔ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ⊆ 𝑥 ) |
| 28 |
26 27
|
sylibr |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑐 ∈ 𝒫 𝑥 ) → ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ∈ 𝒫 𝑥 ) |
| 29 |
28
|
fmpttd |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) : 𝒫 𝑥 ⟶ 𝒫 𝑥 ) |
| 30 |
|
fssxp |
⊢ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) : 𝒫 𝑥 ⟶ 𝒫 𝑥 → ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ⊆ ( 𝒫 𝑥 × 𝒫 𝑥 ) ) |
| 31 |
29 30
|
syl |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ⊆ ( 𝒫 𝑥 × 𝒫 𝑥 ) ) |
| 32 |
|
vpwex |
⊢ 𝒫 𝑥 ∈ V |
| 33 |
32 32
|
xpex |
⊢ ( 𝒫 𝑥 × 𝒫 𝑥 ) ∈ V |
| 34 |
|
ssexg |
⊢ ( ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ⊆ ( 𝒫 𝑥 × 𝒫 𝑥 ) ∧ ( 𝒫 𝑥 × 𝒫 𝑥 ) ∈ V ) → ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ∈ V ) |
| 35 |
31 33 34
|
sylancl |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ∈ V ) |
| 36 |
19
|
adantlr |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → 𝑑 ∈ ( ACS ‘ 𝑥 ) ) |
| 37 |
|
elpwi |
⊢ ( 𝑏 ∈ 𝒫 𝑥 → 𝑏 ⊆ 𝑥 ) |
| 38 |
37
|
ad2antlr |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → 𝑏 ⊆ 𝑥 ) |
| 39 |
21
|
acsfiel2 |
⊢ ( ( 𝑑 ∈ ( ACS ‘ 𝑥 ) ∧ 𝑏 ⊆ 𝑥 ) → ( 𝑏 ∈ 𝑑 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 40 |
36 38 39
|
syl2anc |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → ( 𝑏 ∈ 𝑑 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 41 |
40
|
ralbidva |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( ∀ 𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ↔ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 42 |
|
iunss |
⊢ ( ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) |
| 43 |
42
|
ralbii |
⊢ ( ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) |
| 44 |
|
ralcom |
⊢ ( ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) |
| 45 |
43 44
|
bitri |
⊢ ( ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑑 ∈ 𝑎 ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) |
| 46 |
41 45
|
bitr4di |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( ∀ 𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 47 |
|
elrint2 |
⊢ ( 𝑏 ∈ 𝒫 𝑥 → ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∀ 𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ) ) |
| 48 |
47
|
adantl |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∀ 𝑑 ∈ 𝑎 𝑏 ∈ 𝑑 ) ) |
| 49 |
|
funmpt |
⊢ Fun ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) |
| 50 |
|
funiunfv |
⊢ ( Fun ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ∪ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) = ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ) |
| 51 |
49 50
|
ax-mp |
⊢ ∪ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) = ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) |
| 52 |
51
|
sseq1i |
⊢ ( ∪ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) |
| 53 |
|
iunss |
⊢ ( ∪ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) ⊆ 𝑏 ) |
| 54 |
|
eqid |
⊢ ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) |
| 55 |
|
fveq2 |
⊢ ( 𝑐 = 𝑒 → ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) = ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ) |
| 56 |
55
|
iuneq2d |
⊢ ( 𝑐 = 𝑒 → ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) = ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ) |
| 57 |
|
inss1 |
⊢ ( 𝒫 𝑏 ∩ Fin ) ⊆ 𝒫 𝑏 |
| 58 |
37
|
sspwd |
⊢ ( 𝑏 ∈ 𝒫 𝑥 → 𝒫 𝑏 ⊆ 𝒫 𝑥 ) |
| 59 |
58
|
adantl |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → 𝒫 𝑏 ⊆ 𝒫 𝑥 ) |
| 60 |
57 59
|
sstrid |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( 𝒫 𝑏 ∩ Fin ) ⊆ 𝒫 𝑥 ) |
| 61 |
60
|
sselda |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ) → 𝑒 ∈ 𝒫 𝑥 ) |
| 62 |
20 21
|
mrcssvd |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑑 ∈ 𝑎 ) → ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ) |
| 63 |
62
|
ralrimiva |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ) |
| 64 |
63
|
ad2antrr |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ) → ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ) |
| 65 |
|
iunss |
⊢ ( ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ↔ ∀ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ) |
| 66 |
64 65
|
sylibr |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ) → ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ) |
| 67 |
|
ssexg |
⊢ ( ( ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑥 ∧ 𝑥 ∈ V ) → ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ∈ V ) |
| 68 |
66 11 67
|
sylancl |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ) → ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ∈ V ) |
| 69 |
54 56 61 68
|
fvmptd3 |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ) → ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) = ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ) |
| 70 |
69
|
sseq1d |
⊢ ( ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) ∧ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ) → ( ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 71 |
70
|
ralbidva |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 72 |
53 71
|
syl5bb |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( ∪ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) ‘ 𝑒 ) ⊆ 𝑏 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 73 |
52 72
|
bitr3id |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ↔ ∀ 𝑒 ∈ ( 𝒫 𝑏 ∩ Fin ) ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑒 ) ⊆ 𝑏 ) ) |
| 74 |
46 48 73
|
3bitr4d |
⊢ ( ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) ∧ 𝑏 ∈ 𝒫 𝑥 ) → ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) |
| 75 |
74
|
ralrimiva |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) |
| 76 |
29 75
|
jca |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ) |
| 77 |
|
feq1 |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ↔ ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) : 𝒫 𝑥 ⟶ 𝒫 𝑥 ) ) |
| 78 |
|
imaeq1 |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) = ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ) |
| 79 |
78
|
unieqd |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) = ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ) |
| 80 |
79
|
sseq1d |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ( ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) |
| 81 |
80
|
bibi2d |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ( ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ↔ ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ) |
| 82 |
81
|
ralbidv |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ( ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ↔ ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ) |
| 83 |
77 82
|
anbi12d |
⊢ ( 𝑓 = ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) → ( ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ↔ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( ( 𝑐 ∈ 𝒫 𝑥 ↦ ∪ 𝑑 ∈ 𝑎 ( ( mrCls ‘ 𝑑 ) ‘ 𝑐 ) ) “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ) ) |
| 84 |
35 76 83
|
spcedv |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ∃ 𝑓 ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ) |
| 85 |
|
isacs |
⊢ ( ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ∈ ( ACS ‘ 𝑥 ) ↔ ( ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ∈ ( Moore ‘ 𝑥 ) ∧ ∃ 𝑓 ( 𝑓 : 𝒫 𝑥 ⟶ 𝒫 𝑥 ∧ ∀ 𝑏 ∈ 𝒫 𝑥 ( 𝑏 ∈ ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ↔ ∪ ( 𝑓 “ ( 𝒫 𝑏 ∩ Fin ) ) ⊆ 𝑏 ) ) ) ) |
| 86 |
18 84 85
|
sylanbrc |
⊢ ( 𝑎 ⊆ ( ACS ‘ 𝑥 ) → ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ∈ ( ACS ‘ 𝑥 ) ) |
| 87 |
86
|
adantl |
⊢ ( ( ⊤ ∧ 𝑎 ⊆ ( ACS ‘ 𝑥 ) ) → ( 𝒫 𝑥 ∩ ∩ 𝑎 ) ∈ ( ACS ‘ 𝑥 ) ) |
| 88 |
10 87
|
ismred2 |
⊢ ( ⊤ → ( ACS ‘ 𝑥 ) ∈ ( Moore ‘ 𝒫 𝑥 ) ) |
| 89 |
88
|
mptru |
⊢ ( ACS ‘ 𝑥 ) ∈ ( Moore ‘ 𝒫 𝑥 ) |
| 90 |
4 89
|
vtoclg |
⊢ ( 𝑋 ∈ 𝑉 → ( ACS ‘ 𝑋 ) ∈ ( Moore ‘ 𝒫 𝑋 ) ) |