Step |
Hyp |
Ref |
Expression |
1 |
|
satffunlem2lem1.s |
⊢ 𝑆 = ( 𝑀 Sat 𝐸 ) |
2 |
|
satffunlem2lem1.a |
⊢ 𝐴 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) |
3 |
|
satffunlem2lem1.b |
⊢ 𝐵 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } |
4 |
|
simpl |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → 𝑢 = 𝑠 ) |
5 |
4
|
fveq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) |
6 |
|
simpr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → 𝑣 = 𝑟 ) |
7 |
6
|
fveq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑟 ) ) |
8 |
5 7
|
oveq12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) |
9 |
8
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) ) |
10 |
4
|
fveq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
11 |
6
|
fveq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 2nd ‘ 𝑣 ) = ( 2nd ‘ 𝑟 ) ) |
12 |
10 11
|
ineq12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) = ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) |
13 |
12
|
difeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) |
14 |
2 13
|
syl5eq |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → 𝐴 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) |
15 |
14
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑦 = 𝐴 ↔ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
16 |
9 15
|
anbi12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
17 |
16
|
cbvrexdva |
⊢ ( 𝑢 = 𝑠 → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
18 |
|
simpr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → 𝑖 = 𝑗 ) |
19 |
|
fveq2 |
⊢ ( 𝑢 = 𝑠 → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) |
21 |
18 20
|
goaleq12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ) |
22 |
21
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ) ) |
23 |
3
|
eqeq2i |
⊢ ( 𝑦 = 𝐵 ↔ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) |
24 |
|
opeq1 |
⊢ ( 𝑖 = 𝑗 → 〈 𝑖 , 𝑧 〉 = 〈 𝑗 , 𝑧 〉 ) |
25 |
24
|
sneqd |
⊢ ( 𝑖 = 𝑗 → { 〈 𝑖 , 𝑧 〉 } = { 〈 𝑗 , 𝑧 〉 } ) |
26 |
|
sneq |
⊢ ( 𝑖 = 𝑗 → { 𝑖 } = { 𝑗 } ) |
27 |
26
|
difeq2d |
⊢ ( 𝑖 = 𝑗 → ( ω ∖ { 𝑖 } ) = ( ω ∖ { 𝑗 } ) ) |
28 |
27
|
reseq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) = ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) |
29 |
25 28
|
uneq12d |
⊢ ( 𝑖 = 𝑗 → ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) = ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ) |
30 |
29
|
adantl |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) = ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ) |
31 |
|
fveq2 |
⊢ ( 𝑢 = 𝑠 → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
32 |
31
|
adantr |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
33 |
30 32
|
eleq12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ↔ ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ) ) |
34 |
33
|
ralbidv |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ↔ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ) ) |
35 |
34
|
rabbidv |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) |
36 |
35
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ↔ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) |
37 |
23 36
|
syl5bb |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( 𝑦 = 𝐵 ↔ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) |
38 |
22 37
|
anbi12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑖 = 𝑗 ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) |
39 |
38
|
cbvrexdva |
⊢ ( 𝑢 = 𝑠 → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) |
40 |
17 39
|
orbi12d |
⊢ ( 𝑢 = 𝑠 → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) ) |
41 |
40
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ∃ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ) |
42 |
|
fveq2 |
⊢ ( 𝑣 = 𝑟 → ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑟 ) ) |
43 |
19 42
|
oveqan12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) |
44 |
43
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ) ) |
45 |
2
|
eqeq2i |
⊢ ( 𝑦 = 𝐴 ↔ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) |
46 |
|
fveq2 |
⊢ ( 𝑣 = 𝑟 → ( 2nd ‘ 𝑣 ) = ( 2nd ‘ 𝑟 ) ) |
47 |
31 46
|
ineqan12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) = ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) |
48 |
47
|
difeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) |
49 |
48
|
eqeq2d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ↔ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
50 |
45 49
|
syl5bb |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( 𝑦 = 𝐴 ↔ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
51 |
44 50
|
anbi12d |
⊢ ( ( 𝑢 = 𝑠 ∧ 𝑣 = 𝑟 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
52 |
51
|
cbvrexdva |
⊢ ( 𝑢 = 𝑠 → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
53 |
52
|
cbvrexvw |
⊢ ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
54 |
41 53
|
orbi12i |
⊢ ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ↔ ( ∃ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ∨ ∃ 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ) |
55 |
|
simp-5l |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → Fun ( 𝑆 ‘ suc 𝑁 ) ) |
56 |
|
eldifi |
⊢ ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
57 |
56
|
adantl |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
58 |
57
|
anim1i |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
59 |
58
|
ad2antrr |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
60 |
|
eldifi |
⊢ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
61 |
60
|
adantl |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
62 |
61
|
anim1i |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
63 |
55 59 62
|
3jca |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ) |
64 |
|
id |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
65 |
2
|
eqeq2i |
⊢ ( 𝑤 = 𝐴 ↔ 𝑤 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) |
66 |
65
|
biimpi |
⊢ ( 𝑤 = 𝐴 → 𝑤 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) |
67 |
66
|
anim2i |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
68 |
|
satffunlem |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) → 𝑦 = 𝑤 ) |
69 |
63 64 67 68
|
syl3an |
⊢ ( ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) |
70 |
69
|
3exp |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
71 |
70
|
com23 |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
72 |
71
|
rexlimdva |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
73 |
|
eqeq1 |
⊢ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
74 |
|
fvex |
⊢ ( 1st ‘ 𝑠 ) ∈ V |
75 |
|
fvex |
⊢ ( 1st ‘ 𝑟 ) ∈ V |
76 |
|
gonafv |
⊢ ( ( ( 1st ‘ 𝑠 ) ∈ V ∧ ( 1st ‘ 𝑟 ) ∈ V ) → ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 ) |
77 |
74 75 76
|
mp2an |
⊢ ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 |
78 |
|
df-goal |
⊢ ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 |
79 |
77 78
|
eqeq12i |
⊢ ( ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 = 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 ) |
80 |
|
1oex |
⊢ 1o ∈ V |
81 |
|
opex |
⊢ 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 ∈ V |
82 |
80 81
|
opth |
⊢ ( 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 = 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 ↔ ( 1o = 2o ∧ 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 = 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 ) ) |
83 |
|
1one2o |
⊢ 1o ≠ 2o |
84 |
|
df-ne |
⊢ ( 1o ≠ 2o ↔ ¬ 1o = 2o ) |
85 |
|
pm2.21 |
⊢ ( ¬ 1o = 2o → ( 1o = 2o → 𝑦 = 𝑤 ) ) |
86 |
84 85
|
sylbi |
⊢ ( 1o ≠ 2o → ( 1o = 2o → 𝑦 = 𝑤 ) ) |
87 |
83 86
|
ax-mp |
⊢ ( 1o = 2o → 𝑦 = 𝑤 ) |
88 |
87
|
adantr |
⊢ ( ( 1o = 2o ∧ 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 = 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 ) → 𝑦 = 𝑤 ) |
89 |
82 88
|
sylbi |
⊢ ( 〈 1o , 〈 ( 1st ‘ 𝑠 ) , ( 1st ‘ 𝑟 ) 〉 〉 = 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 → 𝑦 = 𝑤 ) |
90 |
79 89
|
sylbi |
⊢ ( ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → 𝑦 = 𝑤 ) |
91 |
73 90
|
syl6bi |
⊢ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → 𝑦 = 𝑤 ) ) |
92 |
91
|
adantr |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → 𝑦 = 𝑤 ) ) |
93 |
92
|
com12 |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) |
94 |
93
|
adantr |
⊢ ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) |
95 |
94
|
a1i |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
96 |
95
|
rexlimdva |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
97 |
72 96
|
jaod |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
98 |
97
|
rexlimdva |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
99 |
|
simp-4l |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → Fun ( 𝑆 ‘ suc 𝑁 ) ) |
100 |
58
|
adantr |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
101 |
|
ssel |
⊢ ( ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) → ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
102 |
101
|
ad3antlr |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
103 |
102
|
com12 |
⊢ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) → ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
104 |
103
|
adantr |
⊢ ( ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
105 |
104
|
impcom |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
106 |
|
eldifi |
⊢ ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
107 |
106
|
ad2antll |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
108 |
105 107
|
jca |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
109 |
99 100 108
|
3jca |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ) |
110 |
109 64 67 68
|
syl3an |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) |
111 |
110
|
3exp |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
112 |
111
|
com23 |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
113 |
112
|
rexlimdvva |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
114 |
98 113
|
jaod |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → 𝑦 = 𝑤 ) ) ) |
115 |
114
|
com23 |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
116 |
115
|
rexlimdva |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
117 |
|
eqeq1 |
⊢ ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
118 |
|
df-goal |
⊢ ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 |
119 |
|
fvex |
⊢ ( 1st ‘ 𝑢 ) ∈ V |
120 |
|
fvex |
⊢ ( 1st ‘ 𝑣 ) ∈ V |
121 |
|
gonafv |
⊢ ( ( ( 1st ‘ 𝑢 ) ∈ V ∧ ( 1st ‘ 𝑣 ) ∈ V ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 ) |
122 |
119 120 121
|
mp2an |
⊢ ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 |
123 |
118 122
|
eqeq12i |
⊢ ( ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ↔ 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 ) |
124 |
|
2oex |
⊢ 2o ∈ V |
125 |
|
opex |
⊢ 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ∈ V |
126 |
124 125
|
opth |
⊢ ( 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 ↔ ( 2o = 1o ∧ 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 = 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 ) ) |
127 |
87
|
eqcoms |
⊢ ( 2o = 1o → 𝑦 = 𝑤 ) |
128 |
127
|
adantr |
⊢ ( ( 2o = 1o ∧ 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 = 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 ) → 𝑦 = 𝑤 ) |
129 |
126 128
|
sylbi |
⊢ ( 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 = 〈 1o , 〈 ( 1st ‘ 𝑢 ) , ( 1st ‘ 𝑣 ) 〉 〉 → 𝑦 = 𝑤 ) |
130 |
123 129
|
sylbi |
⊢ ( ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → 𝑦 = 𝑤 ) |
131 |
117 130
|
syl6bi |
⊢ ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → 𝑦 = 𝑤 ) ) |
132 |
131
|
adantr |
⊢ ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → 𝑦 = 𝑤 ) ) |
133 |
132
|
com12 |
⊢ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) |
134 |
133
|
adantr |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) |
135 |
134
|
rexlimivw |
⊢ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) |
136 |
135
|
a1i |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
137 |
|
eqeq1 |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ) ) |
138 |
78 118
|
eqeq12i |
⊢ ( ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 ) |
139 |
|
opex |
⊢ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 ∈ V |
140 |
124 139
|
opth |
⊢ ( 〈 2o , 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 〉 = 〈 2o , 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 〉 ↔ ( 2o = 2o ∧ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ) ) |
141 |
|
vex |
⊢ 𝑖 ∈ V |
142 |
141 119
|
opth |
⊢ ( 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ↔ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) |
143 |
142
|
anbi2i |
⊢ ( ( 2o = 2o ∧ 〈 𝑖 , ( 1st ‘ 𝑢 ) 〉 = 〈 𝑗 , ( 1st ‘ 𝑠 ) 〉 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) |
144 |
138 140 143
|
3bitri |
⊢ ( ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) |
145 |
137 144
|
bitrdi |
⊢ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) ) |
146 |
145
|
adantl |
⊢ ( ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ↔ ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) ) ) |
147 |
56
|
a1i |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
148 |
|
funfv1st2nd |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) |
149 |
148
|
ex |
⊢ ( Fun ( 𝑆 ‘ suc 𝑁 ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) ) |
150 |
149
|
adantr |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) ) |
151 |
|
funfv1st2nd |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ) |
152 |
151
|
ex |
⊢ ( Fun ( 𝑆 ‘ suc 𝑁 ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ) ) |
153 |
152
|
adantr |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ) ) |
154 |
|
fveqeq2 |
⊢ ( ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) ↔ ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) ) ) |
155 |
|
eqtr2 |
⊢ ( ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) ∧ ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
156 |
29
|
eqcomd |
⊢ ( 𝑖 = 𝑗 → ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ) |
157 |
156
|
adantl |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) = ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ) |
158 |
|
simpl |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ) |
159 |
158
|
eqcomd |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( 2nd ‘ 𝑠 ) = ( 2nd ‘ 𝑢 ) ) |
160 |
157 159
|
eleq12d |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ↔ ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ) ) |
161 |
160
|
ralbidv |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) ↔ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) ) ) |
162 |
161
|
rabbidv |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) |
163 |
162 3
|
eqtr4di |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } = 𝐵 ) |
164 |
|
eqeq12 |
⊢ ( ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ∧ 𝑤 = 𝐵 ) → ( 𝑦 = 𝑤 ↔ { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } = 𝐵 ) ) |
165 |
163 164
|
syl5ibrcom |
⊢ ( ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) ∧ 𝑖 = 𝑗 ) → ( ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ∧ 𝑤 = 𝐵 ) → 𝑦 = 𝑤 ) ) |
166 |
165
|
exp4b |
⊢ ( ( 2nd ‘ 𝑢 ) = ( 2nd ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
167 |
155 166
|
syl |
⊢ ( ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) ∧ ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
168 |
167
|
ex |
⊢ ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑢 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) |
169 |
154 168
|
syl6bi |
⊢ ( ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) ) |
170 |
169
|
com24 |
⊢ ( ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) → ( 𝑖 = 𝑗 → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) ) |
171 |
170
|
impcom |
⊢ ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) |
172 |
171
|
com13 |
⊢ ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑢 ) ) = ( 2nd ‘ 𝑢 ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) |
173 |
60 153 172
|
syl56 |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) ) |
174 |
173
|
com23 |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ( 𝑆 ‘ suc 𝑁 ) ‘ ( 1st ‘ 𝑠 ) ) = ( 2nd ‘ 𝑠 ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) ) |
175 |
147 150 174
|
3syld |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) ) |
176 |
175
|
imp |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) |
177 |
176
|
adantr |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) ) |
178 |
177
|
imp |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
179 |
178
|
adantld |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
180 |
179
|
ad2antrr |
⊢ ( ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ( 2o = 2o ∧ ( 𝑖 = 𝑗 ∧ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑠 ) ) ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
181 |
146 180
|
sylbid |
⊢ ( ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) → ( 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
182 |
181
|
impd |
⊢ ( ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) ∧ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) |
183 |
182
|
ex |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( 𝑤 = 𝐵 → 𝑦 = 𝑤 ) ) ) ) |
184 |
183
|
com34 |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) → ( 𝑤 = 𝐵 → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) ) |
185 |
184
|
impd |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
186 |
185
|
rexlimdva |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
187 |
136 186
|
jaod |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
188 |
187
|
rexlimdva |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
189 |
134
|
a1i |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
190 |
189
|
rexlimdva |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
191 |
190
|
rexlimdva |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
192 |
188 191
|
jaod |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → 𝑦 = 𝑤 ) ) ) |
193 |
192
|
com23 |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑗 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
194 |
193
|
rexlimdva |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
195 |
116 194
|
jaod |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
196 |
195
|
rexlimdva |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
197 |
|
simplll |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) → Fun ( 𝑆 ‘ suc 𝑁 ) ) |
198 |
|
ssel |
⊢ ( ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) → ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
199 |
198
|
adantrd |
⊢ ( ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) → ( ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
200 |
199
|
adantl |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
201 |
200
|
imp |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
202 |
|
eldifi |
⊢ ( 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
203 |
202
|
ad2antll |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
204 |
201 203
|
jca |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
205 |
204
|
adantr |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
206 |
60
|
anim1i |
⊢ ( ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
207 |
206
|
adantl |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
208 |
197 205 207
|
3jca |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) → ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ) |
209 |
208
|
adantr |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ∧ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ) |
210 |
|
simprl |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ∧ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) |
211 |
67
|
ad2antll |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ∧ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ) |
212 |
209 210 211 68
|
syl3anc |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ∧ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → 𝑦 = 𝑤 ) |
213 |
212
|
exp32 |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
214 |
213
|
impancom |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → ( ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
215 |
214
|
expdimp |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
216 |
215
|
rexlimdv |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) |
217 |
91
|
adantrd |
⊢ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → 𝑦 = 𝑤 ) ) |
218 |
217
|
adantr |
⊢ ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → 𝑦 = 𝑤 ) ) |
219 |
218
|
ad3antlr |
⊢ ( ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑖 ∈ ω ) → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → 𝑦 = 𝑤 ) ) |
220 |
219
|
rexlimdva |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) → 𝑦 = 𝑤 ) ) |
221 |
216 220
|
jaod |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) → 𝑦 = 𝑤 ) ) |
222 |
221
|
rexlimdva |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) → 𝑦 = 𝑤 ) ) |
223 |
|
simplll |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → Fun ( 𝑆 ‘ suc 𝑁 ) ) |
224 |
204
|
adantr |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
225 |
101
|
adantl |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
226 |
225
|
adantr |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
227 |
226
|
com12 |
⊢ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) → ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
228 |
227
|
adantr |
⊢ ( ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
229 |
228
|
impcom |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
230 |
106
|
ad2antll |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) |
231 |
229 230
|
jca |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) |
232 |
223 224 231
|
3jca |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ) ) |
233 |
232 64 67 68
|
syl3an |
⊢ ( ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) |
234 |
233
|
3exp |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
235 |
234
|
impancom |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → ( ( 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) ) |
236 |
235
|
rexlimdvv |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) → 𝑦 = 𝑤 ) ) |
237 |
222 236
|
jaod |
⊢ ( ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) ∧ ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) |
238 |
237
|
ex |
⊢ ( ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∧ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ) → ( ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
239 |
238
|
rexlimdvva |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
240 |
196 239
|
jaod |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑠 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑟 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ∨ ∃ 𝑗 ∈ ω ( 𝑥 = ∀𝑔 𝑗 ( 1st ‘ 𝑠 ) ∧ 𝑦 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑗 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑗 } ) ) ) ∈ ( 2nd ‘ 𝑠 ) } ) ) ∨ ∃ 𝑠 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑟 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑠 ) ⊼𝑔 ( 1st ‘ 𝑟 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑠 ) ∩ ( 2nd ‘ 𝑟 ) ) ) ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
241 |
54 240
|
syl5bi |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) → 𝑦 = 𝑤 ) ) ) |
242 |
241
|
impd |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → 𝑦 = 𝑤 ) ) |
243 |
242
|
alrimivv |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑦 ∀ 𝑤 ( ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → 𝑦 = 𝑤 ) ) |
244 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 = 𝐴 ↔ 𝑤 = 𝐴 ) ) |
245 |
244
|
anbi2d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) |
246 |
245
|
rexbidv |
⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) |
247 |
|
eqeq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 = 𝐵 ↔ 𝑤 = 𝐵 ) ) |
248 |
247
|
anbi2d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ↔ ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ) |
249 |
248
|
rexbidv |
⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ↔ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ) |
250 |
246 249
|
orbi12d |
⊢ ( 𝑦 = 𝑤 → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ) ) |
251 |
250
|
rexbidv |
⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ↔ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ) ) |
252 |
245
|
2rexbidv |
⊢ ( 𝑦 = 𝑤 → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) |
253 |
251 252
|
orbi12d |
⊢ ( 𝑦 = 𝑤 → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ↔ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) ) |
254 |
253
|
mo4 |
⊢ ( ∃* 𝑦 ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ↔ ∀ 𝑦 ∀ 𝑤 ( ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ∧ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑤 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑤 = 𝐴 ) ) ) → 𝑦 = 𝑤 ) ) |
255 |
243 254
|
sylibr |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ∃* 𝑦 ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
256 |
255
|
alrimiv |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑥 ∃* 𝑦 ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
257 |
|
funopab |
⊢ ( Fun { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ↔ ∀ 𝑥 ∃* 𝑦 ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) ) |
258 |
256 257
|
sylibr |
⊢ ( ( Fun ( 𝑆 ‘ suc 𝑁 ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → Fun { 〈 𝑥 , 𝑦 〉 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = 𝐵 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = 𝐴 ) ) } ) |