Step | Hyp | Ref
| Expression |
1 | | simpr 488 |
. . . 4
⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) |
2 | | simpr 488 |
. . . . . . 7
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) |
3 | | peano2 7668 |
. . . . . . . . 9
⊢ (𝑁 ∈ ω → suc 𝑁 ∈
ω) |
4 | 3 | ancri 553 |
. . . . . . . 8
⊢ (𝑁 ∈ ω → (suc
𝑁 ∈ ω ∧
𝑁 ∈
ω)) |
5 | 4 | adantr 484 |
. . . . . . 7
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) |
6 | | sssucid 6290 |
. . . . . . . 8
⊢ 𝑁 ⊆ suc 𝑁 |
7 | 6 | a1i 11 |
. . . . . . 7
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑁 ⊆ suc 𝑁) |
8 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸) |
9 | 8 | satfsschain 33039 |
. . . . . . . 8
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) → (𝑁 ⊆ suc 𝑁 → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁))) |
10 | 9 | imp 410 |
. . . . . . 7
⊢ ((((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω)) ∧ 𝑁 ⊆ suc 𝑁) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) |
11 | 2, 5, 7, 10 | syl21anc 838 |
. . . . . 6
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) |
12 | | eqid 2737 |
. . . . . . . 8
⊢ ((𝑀 ↑m ω)
∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣))) = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣))) |
13 | | eqid 2737 |
. . . . . . . 8
⊢ {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} |
14 | 8, 12, 13 | satffunlem2lem1 33079 |
. . . . . . 7
⊢ ((Fun
((𝑀 Sat 𝐸)‘suc 𝑁) ∧ ((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}) |
15 | 14 | expcom 417 |
. . . . . 6
⊢ (((𝑀 Sat 𝐸)‘𝑁) ⊆ ((𝑀 Sat 𝐸)‘suc 𝑁) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))})) |
16 | 11, 15 | syl 17 |
. . . . 5
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))})) |
17 | 16 | imp 410 |
. . . 4
⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}) |
18 | 8, 12, 13 | satffunlem2lem2 33081 |
. . . 4
⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∩ dom {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}) = ∅) |
19 | | funun 6426 |
. . . 4
⊢ (((Fun
((𝑀 Sat 𝐸)‘suc 𝑁) ∧ Fun {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}) ∧ (dom ((𝑀 Sat 𝐸)‘suc 𝑁) ∩ dom {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}) = ∅) → Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))})) |
20 | 1, 17, 18, 19 | syl21anc 838 |
. . 3
⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))})) |
21 | | simpl 486 |
. . . . . 6
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ 𝑉) |
22 | | simpr 488 |
. . . . . 6
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ 𝑊) |
23 | | simpl 486 |
. . . . . 6
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → 𝑁 ∈ ω) |
24 | 8, 12, 13 | satfvsucsuc 33040 |
. . . . . 6
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘suc suc 𝑁) = (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))})) |
25 | 21, 22, 23, 24 | syl2an23an 1425 |
. . . . 5
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → ((𝑀 Sat 𝐸)‘suc suc 𝑁) = (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))})) |
26 | 25 | funeqd 6402 |
. . . 4
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁) ↔ Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}))) |
27 | 26 | adantr 484 |
. . 3
⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → (Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁) ↔ Fun (((𝑀 Sat 𝐸)‘suc 𝑁) ∪ {〈𝑥, 𝑦〉 ∣ (∃𝑢 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(∃𝑣 ∈ ((𝑀 Sat 𝐸)‘suc 𝑁)(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣
∀𝑗 ∈ 𝑀 ({〈𝑖, 𝑗〉} ∪ (𝑓 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ∨ ∃𝑢 ∈ ((𝑀 Sat 𝐸)‘𝑁)∃𝑣 ∈ (((𝑀 Sat 𝐸)‘suc 𝑁) ∖ ((𝑀 Sat 𝐸)‘𝑁))(𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))}))) |
28 | 20, 27 | mpbird 260 |
. 2
⊢ (((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) ∧ Fun ((𝑀 Sat 𝐸)‘suc 𝑁)) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁)) |
29 | 28 | ex 416 |
1
⊢ ((𝑁 ∈ ω ∧ (𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊)) → (Fun ((𝑀 Sat 𝐸)‘suc 𝑁) → Fun ((𝑀 Sat 𝐸)‘suc suc 𝑁))) |