| Step |
Hyp |
Ref |
Expression |
| 1 |
|
fnmrc |
⊢ mrCls Fn ∪ ran Moore |
| 2 |
|
fnfun |
⊢ ( mrCls Fn ∪ ran Moore → Fun mrCls ) |
| 3 |
1 2
|
ax-mp |
⊢ Fun mrCls |
| 4 |
|
fvelima |
⊢ ( ( Fun mrCls ∧ 𝐹 ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) ) → ∃ 𝑧 ∈ ( Moore ‘ 𝐵 ) ( mrCls ‘ 𝑧 ) = 𝐹 ) |
| 5 |
3 4
|
mpan |
⊢ ( 𝐹 ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) → ∃ 𝑧 ∈ ( Moore ‘ 𝐵 ) ( mrCls ‘ 𝑧 ) = 𝐹 ) |
| 6 |
|
elfvex |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → 𝐵 ∈ V ) |
| 7 |
|
eqid |
⊢ ( mrCls ‘ 𝑧 ) = ( mrCls ‘ 𝑧 ) |
| 8 |
7
|
mrcf |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → ( mrCls ‘ 𝑧 ) : 𝒫 𝐵 ⟶ 𝑧 ) |
| 9 |
|
mresspw |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → 𝑧 ⊆ 𝒫 𝐵 ) |
| 10 |
8 9
|
fssd |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → ( mrCls ‘ 𝑧 ) : 𝒫 𝐵 ⟶ 𝒫 𝐵 ) |
| 11 |
7
|
mrcssid |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) → 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 12 |
11
|
adantrr |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ) → 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 13 |
7
|
mrcss |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ 𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) → ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 14 |
13
|
3expb |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ ( 𝑦 ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵 ) ) → ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 15 |
14
|
ancom2s |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ) → ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 16 |
7
|
mrcidm |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ 𝑥 ⊆ 𝐵 ) → ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 17 |
16
|
adantrr |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ) → ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) |
| 18 |
12 15 17
|
3jca |
⊢ ( ( 𝑧 ∈ ( Moore ‘ 𝐵 ) ∧ ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) |
| 19 |
18
|
ex |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) ) |
| 20 |
19
|
alrimivv |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) ) |
| 21 |
6 10 20
|
3jca |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → ( 𝐵 ∈ V ∧ ( mrCls ‘ 𝑧 ) : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) ) ) |
| 22 |
|
feq1 |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( mrCls ‘ 𝑧 ) : 𝒫 𝐵 ⟶ 𝒫 𝐵 ↔ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ) ) |
| 23 |
|
fveq1 |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) ) |
| 24 |
23
|
sseq2d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ↔ 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
| 25 |
|
fveq1 |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
| 26 |
25 23
|
sseq12d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
| 27 |
|
id |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( mrCls ‘ 𝑧 ) = 𝐹 ) |
| 28 |
27 23
|
fveq12d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) ) |
| 29 |
28 23
|
eqeq12d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
| 30 |
24 26 29
|
3anbi123d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ↔ ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
| 31 |
30
|
imbi2d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) ↔ ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
| 32 |
31
|
2albidv |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
| 33 |
22 32
|
3anbi23d |
⊢ ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( ( 𝐵 ∈ V ∧ ( mrCls ‘ 𝑧 ) : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ 𝑦 ) ⊆ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ∧ ( ( mrCls ‘ 𝑧 ) ‘ ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) = ( ( mrCls ‘ 𝑧 ) ‘ 𝑥 ) ) ) ) ↔ ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) ) |
| 34 |
21 33
|
syl5ibcom |
⊢ ( 𝑧 ∈ ( Moore ‘ 𝐵 ) → ( ( mrCls ‘ 𝑧 ) = 𝐹 → ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) ) |
| 35 |
34
|
rexlimiv |
⊢ ( ∃ 𝑧 ∈ ( Moore ‘ 𝐵 ) ( mrCls ‘ 𝑧 ) = 𝐹 → ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
| 36 |
5 35
|
syl |
⊢ ( 𝐹 ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) → ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
| 37 |
|
simp1 |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → 𝐵 ∈ V ) |
| 38 |
|
simp2 |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ) |
| 39 |
|
ssid |
⊢ 𝑧 ⊆ 𝑧 |
| 40 |
|
3simpb |
⊢ ( ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) |
| 41 |
40
|
imim2i |
⊢ ( ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
| 42 |
41
|
2alimi |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) |
| 43 |
|
sseq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵 ) ) |
| 44 |
43
|
adantr |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( 𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵 ) ) |
| 45 |
|
sseq12 |
⊢ ( ( 𝑦 = 𝑧 ∧ 𝑥 = 𝑧 ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧 ) ) |
| 46 |
45
|
ancoms |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑧 ⊆ 𝑧 ) ) |
| 47 |
44 46
|
anbi12d |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ↔ ( 𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧 ) ) ) |
| 48 |
|
id |
⊢ ( 𝑥 = 𝑧 → 𝑥 = 𝑧 ) |
| 49 |
|
fveq2 |
⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑧 ) ) |
| 50 |
48 49
|
sseq12d |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ↔ 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ) ) |
| 51 |
50
|
adantr |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ↔ 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ) ) |
| 52 |
|
2fveq3 |
⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) ) |
| 53 |
52 49
|
eqeq12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) |
| 54 |
53
|
adantr |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) |
| 55 |
51 54
|
anbi12d |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ↔ ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 56 |
47 55
|
imbi12d |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑧 ) → ( ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ↔ ( ( 𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧 ) → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) ) |
| 57 |
56
|
spc2gv |
⊢ ( ( 𝑧 ∈ V ∧ 𝑧 ∈ V ) → ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧 ) → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) ) |
| 58 |
57
|
el2v |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧 ) → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 59 |
42 58
|
syl |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧 ) → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 60 |
59
|
3ad2ant3 |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑧 ⊆ 𝑧 ) → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 61 |
39 60
|
mpan2i |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → ( 𝑧 ⊆ 𝐵 → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 62 |
61
|
imp |
⊢ ( ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ∧ 𝑧 ⊆ 𝐵 ) → ( 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) ) |
| 63 |
62
|
simpld |
⊢ ( ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ∧ 𝑧 ⊆ 𝐵 ) → 𝑧 ⊆ ( 𝐹 ‘ 𝑧 ) ) |
| 64 |
|
simp2 |
⊢ ( ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) |
| 65 |
64
|
imim2i |
⊢ ( ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
| 66 |
65
|
2alimi |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
| 67 |
66
|
3ad2ant3 |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
| 68 |
43
|
adantr |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝑥 ⊆ 𝐵 ↔ 𝑧 ⊆ 𝐵 ) ) |
| 69 |
|
sseq12 |
⊢ ( ( 𝑦 = 𝑤 ∧ 𝑥 = 𝑧 ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧 ) ) |
| 70 |
69
|
ancoms |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( 𝑦 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑧 ) ) |
| 71 |
68 70
|
anbi12d |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) ↔ ( 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧 ) ) ) |
| 72 |
|
fveq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ) |
| 73 |
|
sseq12 |
⊢ ( ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) ∧ ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑧 ) ) → ( ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) ) |
| 74 |
72 49 73
|
syl2anr |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ↔ ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) ) |
| 75 |
71 74
|
imbi12d |
⊢ ( ( 𝑥 = 𝑧 ∧ 𝑦 = 𝑤 ) → ( ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ↔ ( ( 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧 ) → ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 76 |
75
|
spc2gv |
⊢ ( ( 𝑧 ∈ V ∧ 𝑤 ∈ V ) → ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧 ) → ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) ) ) |
| 77 |
76
|
el2v |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧 ) → ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) ) |
| 78 |
67 77
|
syl |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧 ) → ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) ) |
| 79 |
78
|
3impib |
⊢ ( ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ∧ 𝑧 ⊆ 𝐵 ∧ 𝑤 ⊆ 𝑧 ) → ( 𝐹 ‘ 𝑤 ) ⊆ ( 𝐹 ‘ 𝑧 ) ) |
| 80 |
62
|
simprd |
⊢ ( ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ∧ 𝑧 ⊆ 𝐵 ) → ( 𝐹 ‘ ( 𝐹 ‘ 𝑧 ) ) = ( 𝐹 ‘ 𝑧 ) ) |
| 81 |
37 38 63 79 80
|
ismrcd2 |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → 𝐹 = ( mrCls ‘ dom ( 𝐹 ∩ I ) ) ) |
| 82 |
37 38 63 79 80
|
ismrcd1 |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → dom ( 𝐹 ∩ I ) ∈ ( Moore ‘ 𝐵 ) ) |
| 83 |
|
fvssunirn |
⊢ ( Moore ‘ 𝐵 ) ⊆ ∪ ran Moore |
| 84 |
1
|
fndmi |
⊢ dom mrCls = ∪ ran Moore |
| 85 |
83 84
|
sseqtrri |
⊢ ( Moore ‘ 𝐵 ) ⊆ dom mrCls |
| 86 |
|
funfvima2 |
⊢ ( ( Fun mrCls ∧ ( Moore ‘ 𝐵 ) ⊆ dom mrCls ) → ( dom ( 𝐹 ∩ I ) ∈ ( Moore ‘ 𝐵 ) → ( mrCls ‘ dom ( 𝐹 ∩ I ) ) ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) ) ) |
| 87 |
3 85 86
|
mp2an |
⊢ ( dom ( 𝐹 ∩ I ) ∈ ( Moore ‘ 𝐵 ) → ( mrCls ‘ dom ( 𝐹 ∩ I ) ) ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) ) |
| 88 |
82 87
|
syl |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → ( mrCls ‘ dom ( 𝐹 ∩ I ) ) ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) ) |
| 89 |
81 88
|
eqeltrd |
⊢ ( ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) → 𝐹 ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) ) |
| 90 |
36 89
|
impbii |
⊢ ( 𝐹 ∈ ( mrCls “ ( Moore ‘ 𝐵 ) ) ↔ ( 𝐵 ∈ V ∧ 𝐹 : 𝒫 𝐵 ⟶ 𝒫 𝐵 ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝐵 ∧ 𝑦 ⊆ 𝑥 ) → ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑦 ) ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ ( 𝐹 ‘ 𝑥 ) ) = ( 𝐹 ‘ 𝑥 ) ) ) ) ) |