Step |
Hyp |
Ref |
Expression |
1 |
|
satfsschain.s |
⊢ 𝑆 = ( 𝑀 Sat 𝐸 ) |
2 |
|
fveq2 |
⊢ ( 𝑏 = 𝐵 → ( 𝑆 ‘ 𝑏 ) = ( 𝑆 ‘ 𝐵 ) ) |
3 |
2
|
sseq2d |
⊢ ( 𝑏 = 𝐵 → ( ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ↔ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐵 ) ) ) |
4 |
3
|
imbi2d |
⊢ ( 𝑏 = 𝐵 → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ) ↔ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐵 ) ) ) ) |
5 |
|
fveq2 |
⊢ ( 𝑏 = 𝑎 → ( 𝑆 ‘ 𝑏 ) = ( 𝑆 ‘ 𝑎 ) ) |
6 |
5
|
sseq2d |
⊢ ( 𝑏 = 𝑎 → ( ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ↔ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) ) |
7 |
6
|
imbi2d |
⊢ ( 𝑏 = 𝑎 → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ) ↔ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) ) ) |
8 |
|
fveq2 |
⊢ ( 𝑏 = suc 𝑎 → ( 𝑆 ‘ 𝑏 ) = ( 𝑆 ‘ suc 𝑎 ) ) |
9 |
8
|
sseq2d |
⊢ ( 𝑏 = suc 𝑎 → ( ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ↔ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) ) |
10 |
9
|
imbi2d |
⊢ ( 𝑏 = suc 𝑎 → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ) ↔ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) ) ) |
11 |
|
fveq2 |
⊢ ( 𝑏 = 𝐴 → ( 𝑆 ‘ 𝑏 ) = ( 𝑆 ‘ 𝐴 ) ) |
12 |
11
|
sseq2d |
⊢ ( 𝑏 = 𝐴 → ( ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ↔ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐴 ) ) ) |
13 |
12
|
imbi2d |
⊢ ( 𝑏 = 𝐴 → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑏 ) ) ↔ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐴 ) ) ) ) |
14 |
|
ssidd |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐵 ) ) |
15 |
14
|
a1i |
⊢ ( 𝐵 ∈ ω → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐵 ) ) ) |
16 |
|
pm2.27 |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) ) |
18 |
|
simpr |
⊢ ( ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) |
19 |
|
ssun1 |
⊢ ( 𝑆 ‘ 𝑎 ) ⊆ ( ( 𝑆 ‘ 𝑎 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑎 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑎 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑧 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑧 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) |
20 |
|
simpl |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → 𝑀 ∈ 𝑉 ) |
21 |
|
simpr |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → 𝐸 ∈ 𝑊 ) |
22 |
|
simplll |
⊢ ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝑎 ∈ ω ) |
23 |
1
|
satfvsuc |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑎 ∈ ω ) → ( 𝑆 ‘ suc 𝑎 ) = ( ( 𝑆 ‘ 𝑎 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑎 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑎 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑧 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑧 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
24 |
20 21 22 23
|
syl2an23an |
⊢ ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( 𝑆 ‘ suc 𝑎 ) = ( ( 𝑆 ‘ 𝑎 ) ∪ { 〈 𝑥 , 𝑦 〉 ∣ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑎 ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ 𝑎 ) ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∧ 𝑦 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ) ∨ ∃ 𝑖 ∈ ω ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ∧ 𝑦 = { 𝑧 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑘 ∈ 𝑀 ( { 〈 𝑖 , 𝑘 〉 } ∪ ( 𝑧 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } ) ) } ) ) |
25 |
19 24
|
sseqtrrid |
⊢ ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( 𝑆 ‘ 𝑎 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) |
26 |
25
|
adantr |
⊢ ( ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝑎 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) |
27 |
18 26
|
sstrd |
⊢ ( ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) |
28 |
27
|
ex |
⊢ ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) ) |
29 |
17 28
|
syld |
⊢ ( ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) ) |
30 |
29
|
ex |
⊢ ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) ) ) |
31 |
30
|
com23 |
⊢ ( ( ( 𝑎 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝑎 ) → ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝑎 ) ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ suc 𝑎 ) ) ) ) |
32 |
4 7 10 13 15 31
|
findsg |
⊢ ( ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐴 ) ) ) |
33 |
32
|
ex |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( 𝐵 ⊆ 𝐴 → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐴 ) ) ) ) |
34 |
33
|
com23 |
⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) → ( 𝐵 ⊆ 𝐴 → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐴 ) ) ) ) |
35 |
34
|
impcom |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐵 ⊆ 𝐴 → ( 𝑆 ‘ 𝐵 ) ⊆ ( 𝑆 ‘ 𝐴 ) ) ) |