| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nacsacs |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) → 𝐶 ∈ ( ACS ‘ 𝑋 ) ) |
| 2 |
1
|
acsmred |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) → 𝐶 ∈ ( Moore ‘ 𝑋 ) ) |
| 3 |
|
simpll |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ) |
| 4 |
1
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → 𝐶 ∈ ( ACS ‘ 𝑋 ) ) |
| 5 |
|
elpwi |
⊢ ( 𝑠 ∈ 𝒫 𝐶 → 𝑠 ⊆ 𝐶 ) |
| 6 |
5
|
ad2antlr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → 𝑠 ⊆ 𝐶 ) |
| 7 |
|
simpr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ( toInc ‘ 𝑠 ) ∈ Dirset ) |
| 8 |
|
acsdrsel |
⊢ ( ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ 𝑠 ⊆ 𝐶 ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ∪ 𝑠 ∈ 𝐶 ) |
| 9 |
4 6 7 8
|
syl3anc |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ∪ 𝑠 ∈ 𝐶 ) |
| 10 |
|
eqid |
⊢ ( mrCls ‘ 𝐶 ) = ( mrCls ‘ 𝐶 ) |
| 11 |
10
|
nacsfg |
⊢ ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ ∪ 𝑠 ∈ 𝐶 ) → ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 12 |
3 9 11
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 13 |
10
|
mrefg2 |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ( ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) ) |
| 14 |
2 13
|
syl |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) → ( ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) ) |
| 15 |
14
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ( ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) ) |
| 16 |
12 15
|
mpbid |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ∃ 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 17 |
|
elfpw |
⊢ ( 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) ↔ ( 𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin ) ) |
| 18 |
|
fissuni |
⊢ ( ( 𝑔 ⊆ ∪ 𝑠 ∧ 𝑔 ∈ Fin ) → ∃ ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) 𝑔 ⊆ ∪ ℎ ) |
| 19 |
17 18
|
sylbi |
⊢ ( 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) → ∃ ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) 𝑔 ⊆ ∪ ℎ ) |
| 20 |
|
elfpw |
⊢ ( ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) ↔ ( ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin ) ) |
| 21 |
|
ipodrsfi |
⊢ ( ( ( toInc ‘ 𝑠 ) ∈ Dirset ∧ ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin ) → ∃ 𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖 ) |
| 22 |
21
|
3expb |
⊢ ( ( ( toInc ‘ 𝑠 ) ∈ Dirset ∧ ( ℎ ⊆ 𝑠 ∧ ℎ ∈ Fin ) ) → ∃ 𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖 ) |
| 23 |
20 22
|
sylan2b |
⊢ ( ( ( toInc ‘ 𝑠 ) ∈ Dirset ∧ ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) ) → ∃ 𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖 ) |
| 24 |
|
sstr |
⊢ ( ( 𝑔 ⊆ ∪ ℎ ∧ ∪ ℎ ⊆ 𝑖 ) → 𝑔 ⊆ 𝑖 ) |
| 25 |
24
|
ancoms |
⊢ ( ( ∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ ) → 𝑔 ⊆ 𝑖 ) |
| 26 |
|
simpr |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 27 |
2
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → 𝐶 ∈ ( Moore ‘ 𝑋 ) ) |
| 28 |
|
simprr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → 𝑔 ⊆ 𝑖 ) |
| 29 |
5
|
ad2antlr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → 𝑠 ⊆ 𝐶 ) |
| 30 |
|
simprl |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → 𝑖 ∈ 𝑠 ) |
| 31 |
29 30
|
sseldd |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → 𝑖 ∈ 𝐶 ) |
| 32 |
10
|
mrcsscl |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑔 ⊆ 𝑖 ∧ 𝑖 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ⊆ 𝑖 ) |
| 33 |
27 28 31 32
|
syl3anc |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ⊆ 𝑖 ) |
| 34 |
33
|
adantr |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ⊆ 𝑖 ) |
| 35 |
26 34
|
eqsstrd |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → ∪ 𝑠 ⊆ 𝑖 ) |
| 36 |
|
simplrl |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → 𝑖 ∈ 𝑠 ) |
| 37 |
|
elssuni |
⊢ ( 𝑖 ∈ 𝑠 → 𝑖 ⊆ ∪ 𝑠 ) |
| 38 |
36 37
|
syl |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → 𝑖 ⊆ ∪ 𝑠 ) |
| 39 |
35 38
|
eqssd |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → ∪ 𝑠 = 𝑖 ) |
| 40 |
39 36
|
eqeltrd |
⊢ ( ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) ∧ ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) → ∪ 𝑠 ∈ 𝑠 ) |
| 41 |
40
|
ex |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( 𝑖 ∈ 𝑠 ∧ 𝑔 ⊆ 𝑖 ) ) → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) |
| 42 |
41
|
expr |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ 𝑖 ∈ 𝑠 ) → ( 𝑔 ⊆ 𝑖 → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) |
| 43 |
25 42
|
syl5 |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ 𝑖 ∈ 𝑠 ) → ( ( ∪ ℎ ⊆ 𝑖 ∧ 𝑔 ⊆ ∪ ℎ ) → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) |
| 44 |
43
|
expd |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ 𝑖 ∈ 𝑠 ) → ( ∪ ℎ ⊆ 𝑖 → ( 𝑔 ⊆ ∪ ℎ → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) ) |
| 45 |
44
|
rexlimdva |
⊢ ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) → ( ∃ 𝑖 ∈ 𝑠 ∪ ℎ ⊆ 𝑖 → ( 𝑔 ⊆ ∪ ℎ → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) ) |
| 46 |
23 45
|
syl5 |
⊢ ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) → ( ( ( toInc ‘ 𝑠 ) ∈ Dirset ∧ ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) ) → ( 𝑔 ⊆ ∪ ℎ → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) ) |
| 47 |
46
|
expdimp |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ( ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) → ( 𝑔 ⊆ ∪ ℎ → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) ) |
| 48 |
47
|
rexlimdv |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ( ∃ ℎ ∈ ( 𝒫 𝑠 ∩ Fin ) 𝑔 ⊆ ∪ ℎ → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) |
| 49 |
19 48
|
syl5 |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ( 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) → ( ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) ) |
| 50 |
49
|
rexlimdv |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ( ∃ 𝑔 ∈ ( 𝒫 ∪ 𝑠 ∩ Fin ) ∪ 𝑠 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) → ∪ 𝑠 ∈ 𝑠 ) ) |
| 51 |
16 50
|
mpd |
⊢ ( ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) ∧ ( toInc ‘ 𝑠 ) ∈ Dirset ) → ∪ 𝑠 ∈ 𝑠 ) |
| 52 |
51
|
ex |
⊢ ( ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) → ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) |
| 53 |
52
|
ralrimiva |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) → ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) |
| 54 |
2 53
|
jca |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) → ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ) |
| 55 |
|
simpl |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) → 𝐶 ∈ ( Moore ‘ 𝑋 ) ) |
| 56 |
5
|
adantl |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) → 𝑠 ⊆ 𝐶 ) |
| 57 |
56
|
sseld |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) → ( ∪ 𝑠 ∈ 𝑠 → ∪ 𝑠 ∈ 𝐶 ) ) |
| 58 |
57
|
imim2d |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑠 ∈ 𝒫 𝐶 ) → ( ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) → ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝐶 ) ) ) |
| 59 |
58
|
ralimdva |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ( ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) → ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝐶 ) ) ) |
| 60 |
59
|
imp |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) → ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝐶 ) ) |
| 61 |
|
isacs3 |
⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝐶 ) ) ) |
| 62 |
55 60 61
|
sylanbrc |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) → 𝐶 ∈ ( ACS ‘ 𝑋 ) ) |
| 63 |
10
|
mrcid |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑡 ) = 𝑡 ) |
| 64 |
63
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑡 ) = 𝑡 ) |
| 65 |
62
|
adantr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → 𝐶 ∈ ( ACS ‘ 𝑋 ) ) |
| 66 |
|
mress |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → 𝑡 ⊆ 𝑋 ) |
| 67 |
66
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → 𝑡 ⊆ 𝑋 ) |
| 68 |
65 10 67
|
acsficld |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑡 ) = ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
| 69 |
64 68
|
eqtr3d |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → 𝑡 = ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
| 70 |
10
|
mrcf |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ( mrCls ‘ 𝐶 ) : 𝒫 𝑋 ⟶ 𝐶 ) |
| 71 |
70
|
ffnd |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ( mrCls ‘ 𝐶 ) Fn 𝒫 𝑋 ) |
| 72 |
71
|
adantr |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( mrCls ‘ 𝐶 ) Fn 𝒫 𝑋 ) |
| 73 |
10
|
mrcss |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋 ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ⊆ ( ( mrCls ‘ 𝐶 ) ‘ ℎ ) ) |
| 74 |
73
|
3expb |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ( 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋 ) ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ⊆ ( ( mrCls ‘ 𝐶 ) ‘ ℎ ) ) |
| 75 |
74
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) ∧ ( 𝑔 ⊆ ℎ ∧ ℎ ⊆ 𝑋 ) ) → ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ⊆ ( ( mrCls ‘ 𝐶 ) ‘ ℎ ) ) |
| 76 |
|
vex |
⊢ 𝑡 ∈ V |
| 77 |
|
fpwipodrs |
⊢ ( 𝑡 ∈ V → ( toInc ‘ ( 𝒫 𝑡 ∩ Fin ) ) ∈ Dirset ) |
| 78 |
76 77
|
mp1i |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( toInc ‘ ( 𝒫 𝑡 ∩ Fin ) ) ∈ Dirset ) |
| 79 |
|
inss1 |
⊢ ( 𝒫 𝑡 ∩ Fin ) ⊆ 𝒫 𝑡 |
| 80 |
66
|
sspwd |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → 𝒫 𝑡 ⊆ 𝒫 𝑋 ) |
| 81 |
79 80
|
sstrid |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( 𝒫 𝑡 ∩ Fin ) ⊆ 𝒫 𝑋 ) |
| 82 |
|
fvex |
⊢ ( mrCls ‘ 𝐶 ) ∈ V |
| 83 |
|
imaexg |
⊢ ( ( mrCls ‘ 𝐶 ) ∈ V → ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ V ) |
| 84 |
82 83
|
ax-mp |
⊢ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ V |
| 85 |
84
|
a1i |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ V ) |
| 86 |
72 75 78 81 85
|
ipodrsima |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ∈ Dirset ) |
| 87 |
86
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ∈ Dirset ) |
| 88 |
|
imassrn |
⊢ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ ran ( mrCls ‘ 𝐶 ) |
| 89 |
70
|
frnd |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ran ( mrCls ‘ 𝐶 ) ⊆ 𝐶 ) |
| 90 |
88 89
|
sstrid |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝐶 ) |
| 91 |
90
|
adantr |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝐶 ) |
| 92 |
84
|
elpw |
⊢ ( ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ 𝒫 𝐶 ↔ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ⊆ 𝐶 ) |
| 93 |
91 92
|
sylibr |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ 𝒫 𝐶 ) |
| 94 |
93
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ 𝒫 𝐶 ) |
| 95 |
|
simplr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) |
| 96 |
|
fveq2 |
⊢ ( 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) → ( toInc ‘ 𝑠 ) = ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ) |
| 97 |
96
|
eleq1d |
⊢ ( 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) → ( ( toInc ‘ 𝑠 ) ∈ Dirset ↔ ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ∈ Dirset ) ) |
| 98 |
|
unieq |
⊢ ( 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) → ∪ 𝑠 = ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
| 99 |
|
id |
⊢ ( 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) → 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
| 100 |
98 99
|
eleq12d |
⊢ ( 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) → ( ∪ 𝑠 ∈ 𝑠 ↔ ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ) |
| 101 |
97 100
|
imbi12d |
⊢ ( 𝑠 = ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) → ( ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ↔ ( ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ∈ Dirset → ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ) ) |
| 102 |
101
|
rspcva |
⊢ ( ( ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ 𝒫 𝐶 ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) → ( ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ∈ Dirset → ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ) |
| 103 |
94 95 102
|
syl2anc |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( ( toInc ‘ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ∈ Dirset → ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) ) |
| 104 |
87 103
|
mpd |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ∪ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
| 105 |
69 104
|
eqeltrd |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → 𝑡 ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ) |
| 106 |
|
fvelimab |
⊢ ( ( ( mrCls ‘ 𝐶 ) Fn 𝒫 𝑋 ∧ ( 𝒫 𝑡 ∩ Fin ) ⊆ 𝒫 𝑋 ) → ( 𝑡 ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) = 𝑡 ) ) |
| 107 |
72 81 106
|
syl2anc |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑡 ∈ 𝐶 ) → ( 𝑡 ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) = 𝑡 ) ) |
| 108 |
107
|
adantlr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( 𝑡 ∈ ( ( mrCls ‘ 𝐶 ) “ ( 𝒫 𝑡 ∩ Fin ) ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) = 𝑡 ) ) |
| 109 |
105 108
|
mpbid |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) = 𝑡 ) |
| 110 |
|
eqcom |
⊢ ( 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) = 𝑡 ) |
| 111 |
110
|
rexbii |
⊢ ( ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) = 𝑡 ) |
| 112 |
109 111
|
sylibr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 113 |
10
|
mrefg2 |
⊢ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) → ( ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) ) |
| 114 |
113
|
ad2antrr |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ( ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ↔ ∃ 𝑔 ∈ ( 𝒫 𝑡 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) ) |
| 115 |
112 114
|
mpbird |
⊢ ( ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ∧ 𝑡 ∈ 𝐶 ) → ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 116 |
115
|
ralrimiva |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) → ∀ 𝑡 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) |
| 117 |
10
|
isnacs |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ ∀ 𝑡 ∈ 𝐶 ∃ 𝑔 ∈ ( 𝒫 𝑋 ∩ Fin ) 𝑡 = ( ( mrCls ‘ 𝐶 ) ‘ 𝑔 ) ) ) |
| 118 |
62 116 117
|
sylanbrc |
⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) → 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ) |
| 119 |
54 118
|
impbii |
⊢ ( 𝐶 ∈ ( NoeACS ‘ 𝑋 ) ↔ ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝑠 ) ) ) |