Step |
Hyp |
Ref |
Expression |
1 |
|
fveq2 |
⊢ ( 𝑢 = 𝑠 → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) |
2 |
|
fveq2 |
⊢ ( 𝑣 = 𝑟 → ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑟 ) ) |
3 |
1 2
|
oveqan12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) |
4 |
3
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) ) |
5 |
|
fveq2 |
⊢ ( 𝑢 = 𝑠 → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
6 |
|
fveq2 |
⊢ ( 𝑣 = 𝑟 → ( 2nd ‘ 𝑣 ) = ( 2nd ‘ 𝑟 ) ) |
7 |
5 6
|
ineqan12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) = ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) |
8 |
7
|
difeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) |
9 |
8
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ↔ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
10 |
4 9
|
anbi12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
11 |
10
|
cbvrexdva |
⊢ ( 𝑢 = 𝑠 → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ↔ ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
12 |
|
simpr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → 𝑖 = 𝑗 ) |
13 |
1
|
adantr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) |
14 |
12 13
|
goaleq12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ) |
15 |
14
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ) ) |
16 |
|
opeq1 |
⊢ ( 𝑖 = 𝑗 → 〈 𝑖 , 𝑘 〉 = 〈 𝑗 , 𝑘 〉 ) |
17 |
16
|
sneqd |
⊢ ( 𝑖 = 𝑗 → { 〈 𝑖 , 𝑘 〉 } = { 〈 𝑗 , 𝑘 〉 } ) |
18 |
|
sneq |
⊢ ( 𝑖 = 𝑗 → { 𝑖 } = { 𝑗 } ) |
19 |
18
|
difeq2d |
⊢ ( 𝑖 = 𝑗 → ( ω ∖ { 𝑖 } ) = ( ω ∖ { 𝑗 } ) ) |
20 |
19
|
reseq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) = ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) |
21 |
17 20
|
uneq12d |
⊢ ( 𝑖 = 𝑗 → ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) = ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ) |
22 |
21
|
adantl |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) = ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ) |
23 |
5
|
adantr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
24 |
22 23
|
eleq12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ↔ ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ) ) |
25 |
24
|
ralbidv |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ↔ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ) ) |
26 |
25
|
rabbidv |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) |
27 |
26
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ↔ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) |
28 |
15 27
|
anbi12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ↔ ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) |
29 |
28
|
cbvrexdva |
⊢ ( 𝑢 = 𝑠 → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ↔ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) |
30 |
11 29
|
orbi12d |
⊢ ( 𝑢 = 𝑠 → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) ) |
31 |
30
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ∃ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) |
32 |
|
simp-4l |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
33 |
|
simpr |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
34 |
33
|
anim1i |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
35 |
|
simpr |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
36 |
35
|
anim1i |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
37 |
36
|
ad2antrr |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
38 |
|
satffunlem |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → 𝑧 = 𝑦 ) |
39 |
38
|
eqcomd |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → 𝑦 = 𝑧 ) |
40 |
39
|
3exp |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
41 |
32 34 37 40
|
syl3anc |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
42 |
41
|
rexlimdva |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
43 |
|
eqeq1 |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ↔ ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) ) |
44 |
|
df-goal |
⊢ ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 |
45 |
|
fvex |
⊢ ( 1st ‘ 𝑠 ) ∈ V |
46 |
|
fvex |
⊢ ( 1st ‘ 𝑟 ) ∈ V |
47 |
|
gonafv |
⊢ ( ( ( 1st ‘ 𝑠 ) ∈ V ∧ ( 1st ‘ 𝑟 ) ∈ V ) → ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 ) |
48 |
45 46 47
|
mp2an |
⊢ ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 |
49 |
44 48
|
eqeq12i |
⊢ ( ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ↔ 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 ) |
50 |
|
2oex |
⊢ 2o ∈ V |
51 |
|
opex |
⊢ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 ∈ V |
52 |
50 51
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 ↔ ( 2o = 1o ∧ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 ) ) |
53 |
|
1one2o |
⊢ 1o ≠ 2o |
54 |
|
df-ne |
⊢ ( 1o ≠ 2o ↔ ¬ 1o = 2o ) |
55 |
|
pm2.21 |
⊢ ( ¬ 1o = 2o → ( 1o = 2o → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) ) |
56 |
54 55
|
sylbi |
⊢ ( 1o ≠ 2o → ( 1o = 2o → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) ) |
57 |
53 56
|
ax-mp |
⊢ ( 1o = 2o → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) |
58 |
57
|
eqcoms |
⊢ ( 2o = 1o → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) |
59 |
58
|
adantr |
⊢ ( ( 2o = 1o ∧ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 ) → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) |
60 |
52 59
|
sylbi |
⊢ ( 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) |
61 |
49 60
|
sylbi |
⊢ ( ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) |
62 |
43 61
|
syl6bi |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) → 𝑦 = 𝑧 ) ) ) |
63 |
62
|
impd |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) |
64 |
63
|
adantr |
⊢ ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) |
65 |
64
|
a1i |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
66 |
65
|
rexlimdva |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
67 |
42 66
|
jaod |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
68 |
67
|
rexlimdva |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑧 ) ) ) |
69 |
68
|
com23 |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
70 |
69
|
rexlimdva |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
71 |
|
eqeq1 |
⊢ ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
72 |
|
df-goal |
⊢ ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 |
73 |
|
fvex |
⊢ ( 1st ‘ 𝑢 ) ∈ V |
74 |
|
fvex |
⊢ ( 1st ‘ 𝑣 ) ∈ V |
75 |
|
gonafv |
⊢ ( ( ( 1st ‘ 𝑢 ) ∈ V ∧ ( 1st ‘ 𝑣 ) ∈ V ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 ) |
76 |
73 74 75
|
mp2an |
⊢ ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 |
77 |
72 76
|
eqeq12i |
⊢ ( ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 ) |
78 |
|
opex |
⊢ 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ∈ V |
79 |
50 78
|
opth |
⊢ ( 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 ↔ ( 2o = 1o ∧ 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 = 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 ) ) |
80 |
|
pm2.21 |
⊢ ( ¬ 1o = 2o → ( 1o = 2o → 𝑦 = 𝑧 ) ) |
81 |
54 80
|
sylbi |
⊢ ( 1o ≠ 2o → ( 1o = 2o → 𝑦 = 𝑧 ) ) |
82 |
53 81
|
ax-mp |
⊢ ( 1o = 2o → 𝑦 = 𝑧 ) |
83 |
82
|
eqcoms |
⊢ ( 2o = 1o → 𝑦 = 𝑧 ) |
84 |
83
|
adantr |
⊢ ( ( 2o = 1o ∧ 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 = 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 ) → 𝑦 = 𝑧 ) |
85 |
79 84
|
sylbi |
⊢ ( 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 → 𝑦 = 𝑧 ) |
86 |
77 85
|
sylbi |
⊢ ( ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → 𝑦 = 𝑧 ) |
87 |
71 86
|
syl6bi |
⊢ ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → 𝑦 = 𝑧 ) ) |
88 |
87
|
adantr |
⊢ ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → 𝑦 = 𝑧 ) ) |
89 |
88
|
com12 |
⊢ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) |
90 |
89
|
adantr |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) |
91 |
90
|
a1i |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) |
92 |
91
|
rexlimdva |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) |
93 |
|
eqeq1 |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ) ) |
94 |
44 72
|
eqeq12i |
⊢ ( ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 ) |
95 |
50 51
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 ↔ ( 2o = 2o ∧ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ) ) |
96 |
|
vex |
⊢ 𝑖 ∈ V |
97 |
96 73
|
opth |
⊢ ( 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ↔ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) |
98 |
97
|
anbi2i |
⊢ ( ( 2o = 2o ∧ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) |
99 |
94 95 98
|
3bitri |
⊢ ( ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) |
100 |
93 99
|
bitrdi |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) ) |
101 |
100
|
adantl |
⊢ ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) ) |
102 |
|
funfv1st2nd |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) |
103 |
102
|
ex |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) ) |
104 |
|
funfv1st2nd |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ) |
105 |
104
|
ex |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ) ) |
106 |
|
fveqeq2 |
⊢ ( ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ↔ ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) ) ) |
107 |
|
eqtr2 |
⊢ ( ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) ∧ ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
108 |
|
opeq1 |
⊢ ( 𝑗 = 𝑖 → 〈 𝑗 , 𝑘 〉 = 〈 𝑖 , 𝑘 〉 ) |
109 |
108
|
sneqd |
⊢ ( 𝑗 = 𝑖 → { 〈 𝑗 , 𝑘 〉 } = { 〈 𝑖 , 𝑘 〉 } ) |
110 |
|
sneq |
⊢ ( 𝑗 = 𝑖 → { 𝑗 } = { 𝑖 } ) |
111 |
110
|
difeq2d |
⊢ ( 𝑗 = 𝑖 → ( ω ∖ { 𝑗 } ) = ( ω ∖ { 𝑖 } ) ) |
112 |
111
|
reseq2d |
⊢ ( 𝑗 = 𝑖 → ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) = ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) |
113 |
109 112
|
uneq12d |
⊢ ( 𝑗 = 𝑖 → ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ) |
114 |
113
|
eqcoms |
⊢ ( 𝑖 = 𝑗 → ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ) |
115 |
114
|
adantl |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ) |
116 |
|
simpl |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
117 |
116
|
eqcomd |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( 2nd ‘ 𝑠 ) = ( 2nd ‘ 𝑢 ) ) |
118 |
115 117
|
eleq12d |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ↔ ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ) ) |
119 |
118
|
ralbidv |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ↔ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ) ) |
120 |
119
|
rabbidv |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) |
121 |
|
eqeq12 |
⊢ ( ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → ( 𝑦 = 𝑧 ↔ { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) |
122 |
120 121
|
syl5ibrcom |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → 𝑦 = 𝑧 ) ) |
123 |
122
|
exp4b |
⊢ ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
124 |
107 123
|
syl |
⊢ ( ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) ∧ ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
125 |
124
|
ex |
⊢ ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) |
126 |
106 125
|
syl6bi |
⊢ ( ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) ) |
127 |
126
|
com24 |
⊢ ( ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) ) |
128 |
127
|
impcom |
⊢ ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) |
129 |
128
|
com13 |
⊢ ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) |
130 |
105 129
|
syl6 |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) ) |
131 |
130
|
com23 |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) ) |
132 |
103 131
|
syld |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) ) |
133 |
132
|
imp |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) |
134 |
133
|
adantr |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) ) |
135 |
134
|
imp |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
136 |
135
|
adantld |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
137 |
136
|
ad2antrr |
⊢ ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
138 |
101 137
|
sylbid |
⊢ ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
139 |
138
|
impd |
⊢ ( ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) |
140 |
139
|
ex |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → 𝑦 = 𝑧 ) ) ) ) |
141 |
140
|
com34 |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) ) |
142 |
141
|
impd |
⊢ ( ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) |
143 |
142
|
rexlimdva |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) |
144 |
92 143
|
jaod |
⊢ ( ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) |
145 |
144
|
rexlimdva |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑧 ) ) ) |
146 |
145
|
com23 |
⊢ ( ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ∧ 𝑗 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
147 |
146
|
rexlimdva |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
148 |
70 147
|
jaod |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ∧ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
149 |
148
|
rexlimdva |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ∃ 𝑠 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑟 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑗 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
150 |
31 149
|
syl5bi |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) → 𝑦 = 𝑧 ) ) ) |
151 |
150
|
impd |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ∧ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) → 𝑦 = 𝑧 ) ) |
152 |
151
|
alrimivv |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ∀ 𝑦 ∀ 𝑧 ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ∧ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) → 𝑦 = 𝑧 ) ) |
153 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ↔ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
154 |
153
|
anbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) ) |
155 |
154
|
rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) ) |
156 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ↔ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) |
157 |
156
|
anbi2d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ↔ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
158 |
157
|
rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ↔ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
159 |
155 158
|
orbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
160 |
159
|
rexbidv |
⊢ ( 𝑦 = 𝑧 → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) ) |
161 |
160
|
mo4 |
⊢ ( ∃* 𝑦 ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ∧ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑧 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑧 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) → 𝑦 = 𝑧 ) ) |
162 |
152 161
|
sylibr |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ∃* 𝑦 ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
163 |
162
|
alrimiv |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ∀ 𝑥 ∃* 𝑦 ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
164 |
|
funopab |
⊢ ( Fun { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ↔ ∀ 𝑥 ∃* 𝑦 ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) ) |
165 |
163 164
|
sylibr |
⊢ ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → Fun { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑓 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑓 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) |