Step |
Hyp |
Ref |
Expression |
1 |
|
satffunlem2lem2.s |
⊢ 𝑆 = ( 𝑀 Sat 𝐸 ) |
2 |
|
satffunlem2lem2.a |
⊢ 𝐴 = ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) |
3 |
|
satffunlem2lem2.b |
⊢ 𝐵 = { 𝑎 ∈ ( 𝑀 ↑m ω ) ∣ ∀ 𝑧 ∈ 𝑀 ( { 〈 𝑖 , 𝑧 〉 } ∪ ( 𝑎 ↾ ( ω ∖ { 𝑖 } ) ) ) ∈ ( 2nd ‘ 𝑢 ) } |
4 |
1
|
fveq1i |
⊢ ( 𝑆 ‘ suc 𝑁 ) = ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) |
5 |
4
|
dmeqi |
⊢ dom ( 𝑆 ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) |
6 |
|
simprl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝑀 ∈ 𝑉 ) |
7 |
|
simprr |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝐸 ∈ 𝑊 ) |
8 |
|
peano2 |
⊢ ( 𝑁 ∈ ω → suc 𝑁 ∈ ω ) |
9 |
8
|
adantr |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → suc 𝑁 ∈ ω ) |
10 |
6 7 9
|
3jca |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) ) |
11 |
|
satfdmfmla |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) ) |
12 |
10 11
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) ) |
13 |
12
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) ) |
14 |
5 13
|
syl5eq |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( 𝑆 ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) ) |
15 |
|
ovex |
⊢ ( 𝑀 ↑m ω ) ∈ V |
16 |
15
|
difexi |
⊢ ( ( 𝑀 ↑m ω ) ∖ ( ( 2nd ‘ 𝑢 ) ∩ ( 2nd ‘ 𝑣 ) ) ) ∈ V |
17 |
2 16
|
eqeltri |
⊢ 𝐴 ∈ V |
18 |
17
|
a1i |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → 𝐴 ∈ V ) |
19 |
18
|
ralrimiva |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ) |
20 |
3 15
|
rabex2 |
⊢ 𝐵 ∈ V |
21 |
20
|
a1i |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑖 ∈ ω ) → 𝐵 ∈ V ) |
22 |
21
|
ralrimiva |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑖 ∈ ω 𝐵 ∈ V ) |
23 |
19 22
|
jca |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ∧ ∀ 𝑖 ∈ ω 𝐵 ∈ V ) ) |
24 |
23
|
ralrimiva |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ∀ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ∧ ∀ 𝑖 ∈ ω 𝐵 ∈ V ) ) |
25 |
|
simplr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) |
26 |
8
|
ancri |
⊢ ( 𝑁 ∈ ω → ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) |
27 |
26
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) |
28 |
25 27
|
jca |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) ) |
29 |
|
sssucid |
⊢ 𝑁 ⊆ suc 𝑁 |
30 |
1
|
satfsschain |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) → ( 𝑁 ⊆ suc 𝑁 → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) ) |
31 |
28 29 30
|
mpisyl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) |
32 |
|
dmopab3rexdif |
⊢ ( ( ∀ 𝑢 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ∀ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝐴 ∈ V ∧ ∀ 𝑖 ∈ ω 𝐵 ∈ V ) ∧ ( 𝑆 ‘ 𝑁 ) ⊆ ( 𝑆 ‘ suc 𝑁 ) ) → dom { 〈 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 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) } ) |
33 |
24 31 32
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom { 〈 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 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) } ) |
34 |
|
simpr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
35 |
|
fveqeq2 |
⊢ ( 𝑤 = 𝑢 → ( ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ↔ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑢 ) ) ) |
36 |
35
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ 𝑤 = 𝑢 ) → ( ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ↔ ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑢 ) ) ) |
37 |
|
eqidd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑢 ) = ( 1st ‘ 𝑢 ) ) |
38 |
34 36 37
|
rspcedvd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ∃ 𝑤 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ) |
39 |
4
|
funeqi |
⊢ ( Fun ( 𝑆 ‘ suc 𝑁 ) ↔ Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
40 |
39
|
biimpi |
⊢ ( Fun ( 𝑆 ‘ suc 𝑁 ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
41 |
40
|
adantl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
42 |
1
|
fveq1i |
⊢ ( 𝑆 ‘ 𝑁 ) = ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) |
43 |
31 42 4
|
3sstr3g |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
44 |
41 43
|
jca |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) ) |
45 |
44
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) ) |
46 |
|
funeldmdif |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑤 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ) ) |
47 |
45 46
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑤 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑤 ) = ( 1st ‘ 𝑢 ) ) ) |
48 |
38 47
|
mpbird |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
49 |
48
|
ex |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
50 |
4 42
|
difeq12i |
⊢ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) = ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
51 |
50
|
eleq2i |
⊢ ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
52 |
51
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
53 |
13
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
54 |
|
simpl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → 𝑁 ∈ ω ) |
55 |
6 7 54
|
3jca |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ) |
56 |
|
satfdmfmla |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) ) |
57 |
55 56
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) ) |
58 |
57
|
eqcomd |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( Fmla ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
59 |
58
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
60 |
53 59
|
difeq12d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) = ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
61 |
60
|
eleq2d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ ( 1st ‘ 𝑢 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
62 |
49 52 61
|
3imtr4d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) |
63 |
62
|
imp |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
64 |
63
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
65 |
|
oveq1 |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( 𝑓 ⊼𝑔 𝑔 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) |
66 |
65
|
eqeq2d |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
67 |
66
|
rexbidv |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
68 |
|
eqidd |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → 𝑖 = 𝑖 ) |
69 |
|
id |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → 𝑓 = ( 1st ‘ 𝑢 ) ) |
70 |
68 69
|
goaleq12d |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ∀𝑔 𝑖 𝑓 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) |
71 |
70
|
eqeq2d |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( 𝑥 = ∀𝑔 𝑖 𝑓 ↔ 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
72 |
71
|
rexbidv |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ↔ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
73 |
67 72
|
orbi12d |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
74 |
73
|
adantl |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ∧ 𝑓 = ( 1st ‘ 𝑢 ) ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
75 |
6
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → 𝑀 ∈ 𝑉 ) |
76 |
7
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → 𝐸 ∈ 𝑊 ) |
77 |
8
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → suc 𝑁 ∈ ω ) |
78 |
75 76 77
|
3jca |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) ) |
79 |
|
satfrel |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
80 |
78 79
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
81 |
4
|
releqi |
⊢ ( Rel ( 𝑆 ‘ suc 𝑁 ) ↔ Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
82 |
80 81
|
sylibr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( 𝑆 ‘ suc 𝑁 ) ) |
83 |
|
1stdm |
⊢ ( ( Rel ( 𝑆 ‘ suc 𝑁 ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ dom ( 𝑆 ‘ suc 𝑁 ) ) |
84 |
82 83
|
sylan |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ dom ( 𝑆 ‘ suc 𝑁 ) ) |
85 |
14
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( 𝑆 ‘ suc 𝑁 ) ) |
86 |
85
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( 𝑆 ‘ suc 𝑁 ) ) |
87 |
84 86
|
eleqtrrd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( Fmla ‘ suc 𝑁 ) ) |
88 |
87
|
ad4ant13 |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( Fmla ‘ suc 𝑁 ) ) |
89 |
|
oveq2 |
⊢ ( 𝑔 = ( 1st ‘ 𝑣 ) → ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) |
90 |
89
|
eqeq2d |
⊢ ( 𝑔 = ( 1st ‘ 𝑣 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
91 |
90
|
adantl |
⊢ ( ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ∧ 𝑔 = ( 1st ‘ 𝑣 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
92 |
|
simpr |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) |
93 |
88 91 92
|
rspcedvd |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) |
94 |
93
|
rexlimdva2 |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
95 |
94
|
orim1d |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
96 |
95
|
imp |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) |
97 |
64 74 96
|
rspcedvd |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) → ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) |
98 |
97
|
rexlimdva2 |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) ) |
99 |
55
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) ) |
100 |
|
satfrel |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
101 |
99 100
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
102 |
42
|
releqi |
⊢ ( Rel ( 𝑆 ‘ 𝑁 ) ↔ Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
103 |
101 102
|
sylibr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( 𝑆 ‘ 𝑁 ) ) |
104 |
|
1stdm |
⊢ ( ( Rel ( 𝑆 ‘ 𝑁 ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ dom ( 𝑆 ‘ 𝑁 ) ) |
105 |
103 104
|
sylan |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ dom ( 𝑆 ‘ 𝑁 ) ) |
106 |
42
|
dmeqi |
⊢ dom ( 𝑆 ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) |
107 |
99 56
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) ) |
108 |
106 107
|
syl5eq |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( 𝑆 ‘ 𝑁 ) = ( Fmla ‘ 𝑁 ) ) |
109 |
108
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( 𝑆 ‘ 𝑁 ) ) |
110 |
109
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( 𝑆 ‘ 𝑁 ) ) |
111 |
105 110
|
eleqtrrd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑢 ) ∈ ( Fmla ‘ 𝑁 ) ) |
112 |
111
|
adantr |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( 1st ‘ 𝑢 ) ∈ ( Fmla ‘ 𝑁 ) ) |
113 |
66
|
rexbidv |
⊢ ( 𝑓 = ( 1st ‘ 𝑢 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
114 |
113
|
adantl |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ∧ 𝑓 = ( 1st ‘ 𝑢 ) ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
115 |
|
simpr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
116 |
|
fveqeq2 |
⊢ ( 𝑡 = 𝑣 → ( ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ↔ ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑣 ) ) ) |
117 |
116
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ∧ 𝑡 = 𝑣 ) → ( ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ↔ ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑣 ) ) ) |
118 |
|
eqidd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑣 ) = ( 1st ‘ 𝑣 ) ) |
119 |
115 117 118
|
rspcedvd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ∃ 𝑡 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ) |
120 |
44
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) ) |
121 |
|
funeldmdif |
⊢ ( ( Fun ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑡 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ) ) |
122 |
120 121
|
syl |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ↔ ∃ 𝑡 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑡 ) = ( 1st ‘ 𝑣 ) ) ) |
123 |
119 122
|
mpbird |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
124 |
123
|
ex |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
125 |
50
|
eleq2i |
⊢ ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
126 |
125
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ↔ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
127 |
12
|
eqcomd |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
128 |
127 58
|
difeq12d |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) = ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
129 |
128
|
eleq2d |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
130 |
129
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ ( 1st ‘ 𝑣 ) ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
131 |
124 126 130
|
3imtr4d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) |
132 |
131
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) → ( 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) ) |
133 |
132
|
imp |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
134 |
133
|
adantr |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( 1st ‘ 𝑣 ) ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ) |
135 |
90
|
adantl |
⊢ ( ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ∧ 𝑔 = ( 1st ‘ 𝑣 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
136 |
|
simpr |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) |
137 |
134 135 136
|
rspcedvd |
⊢ ( ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) |
138 |
137
|
r19.29an |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) |
139 |
112 114 138
|
rspcedvd |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ) ∧ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) |
140 |
139
|
rexlimdva2 |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) → ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
141 |
98 140
|
orim12d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) → ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) ) |
142 |
10
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) ) |
143 |
11
|
eqcomd |
⊢ ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ suc 𝑁 ∈ ω ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
144 |
142 143
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
145 |
107
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
146 |
144 145
|
difeq12d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) = ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
147 |
146
|
eleq2d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ 𝑓 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
148 |
|
eqid |
⊢ ( 𝑀 Sat 𝐸 ) = ( 𝑀 Sat 𝐸 ) |
149 |
148
|
satfsschain |
⊢ ( ( ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ∧ ( suc 𝑁 ∈ ω ∧ 𝑁 ∈ ω ) ) → ( 𝑁 ⊆ suc 𝑁 → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) ) |
150 |
28 29 149
|
mpisyl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
151 |
|
releldmdifi |
⊢ ( ( Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) ) |
152 |
80 150 151
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) ) |
153 |
147 152
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) ) |
154 |
50
|
eqcomi |
⊢ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) = ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) |
155 |
154
|
rexeqi |
⊢ ( ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ↔ ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ) |
156 |
|
r19.41v |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) ↔ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) ) |
157 |
|
oveq1 |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) = ( 𝑓 ⊼𝑔 𝑔 ) ) |
158 |
157
|
eqeq2d |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
159 |
158
|
rexbidv |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
160 |
|
eqidd |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → 𝑖 = 𝑖 ) |
161 |
|
id |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( 1st ‘ 𝑢 ) = 𝑓 ) |
162 |
160 161
|
goaleq12d |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) = ∀𝑔 𝑖 𝑓 ) |
163 |
162
|
eqeq2d |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ 𝑥 = ∀𝑔 𝑖 𝑓 ) ) |
164 |
163
|
rexbidv |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ↔ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) |
165 |
159 164
|
orbi12d |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) ) |
166 |
165
|
adantl |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ↔ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) ) |
167 |
142 11
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( Fmla ‘ suc 𝑁 ) ) |
168 |
167
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( Fmla ‘ suc 𝑁 ) = dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) |
169 |
168
|
eleq2d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) ↔ 𝑔 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) ) |
170 |
|
releldm2 |
⊢ ( Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) → ( 𝑔 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ) ) |
171 |
80 170
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ) ) |
172 |
169 171
|
bitrd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) ↔ ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ) ) |
173 |
|
r19.41v |
⊢ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ↔ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
174 |
1
|
eqcomi |
⊢ ( 𝑀 Sat 𝐸 ) = 𝑆 |
175 |
174
|
fveq1i |
⊢ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) = ( 𝑆 ‘ suc 𝑁 ) |
176 |
175
|
rexeqi |
⊢ ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ↔ ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) ) |
177 |
89
|
eqcoms |
⊢ ( ( 1st ‘ 𝑣 ) = 𝑔 → ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) |
178 |
177
|
eqeq2d |
⊢ ( ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
179 |
178
|
biimpa |
⊢ ( ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) |
180 |
179
|
a1i |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
181 |
180
|
reximdv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
182 |
176 181
|
syl5bi |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
183 |
173 182
|
syl5bir |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
184 |
183
|
expd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
185 |
172 184
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
186 |
185
|
rexlimdv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
187 |
186
|
ad2antrr |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
188 |
187
|
orim1d |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
189 |
166 188
|
sylbird |
⊢ ( ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
190 |
189
|
expimpd |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ) → ( ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) → ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
191 |
190
|
reximdva |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
192 |
156 191
|
syl5bir |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
193 |
192
|
expd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
194 |
155 193
|
syl5bi |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑢 ) = 𝑓 → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
195 |
153 194
|
syld |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ( ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) ) |
196 |
195
|
rexlimdv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) → ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ) ) |
197 |
145
|
eleq2d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ↔ 𝑓 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) |
198 |
55 100
|
syl |
⊢ ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
199 |
198
|
adantr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) |
200 |
|
releldm2 |
⊢ ( Rel ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) → ( 𝑓 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ) ) |
201 |
199 200
|
syl |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ) ) |
202 |
197 201
|
bitrd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) ↔ ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ) ) |
203 |
|
r19.41v |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ↔ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
204 |
42
|
eqcomi |
⊢ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) = ( 𝑆 ‘ 𝑁 ) |
205 |
204
|
rexeqi |
⊢ ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ↔ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
206 |
158
|
rexbidv |
⊢ ( ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
207 |
206
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ↔ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) |
208 |
146
|
eleq2d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ↔ 𝑔 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ) ) |
209 |
|
releldmdifi |
⊢ ( ( Rel ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∧ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ⊆ ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) ) |
210 |
80 150 209
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( dom ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ dom ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) → ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) ) |
211 |
208 210
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) ) |
212 |
154
|
rexeqi |
⊢ ( ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ↔ ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 ) |
213 |
178
|
biimpcd |
⊢ ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ( ( 1st ‘ 𝑣 ) = 𝑔 → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
214 |
213
|
adantl |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ( ( 1st ‘ 𝑣 ) = 𝑔 → 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
215 |
214
|
reximdv |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
216 |
215
|
ex |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
217 |
216
|
com23 |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
218 |
212 217
|
syl5bi |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑣 ∈ ( ( ( 𝑀 Sat 𝐸 ) ‘ suc 𝑁 ) ∖ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ) ( 1st ‘ 𝑣 ) = 𝑔 → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
219 |
211 218
|
syld |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) → ( 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
220 |
219
|
rexlimdv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
221 |
220
|
adantr |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
222 |
207 221
|
sylbird |
⊢ ( ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) ∧ ( 1st ‘ 𝑢 ) = 𝑓 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
223 |
222
|
expimpd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
224 |
223
|
reximdv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
225 |
205 224
|
syl5bi |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
226 |
203 225
|
syl5bir |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 ∧ ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
227 |
226
|
expd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑀 Sat 𝐸 ) ‘ 𝑁 ) ( 1st ‘ 𝑢 ) = 𝑓 → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
228 |
202 227
|
sylbid |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( 𝑓 ∈ ( Fmla ‘ 𝑁 ) → ( ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
229 |
228
|
rexlimdv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) → ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) |
230 |
196 229
|
orim12d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) → ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ) ) |
231 |
141 230
|
impbid |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) ↔ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) ) ) |
232 |
231
|
abbidv |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → { 𝑥 ∣ ( ∃ 𝑢 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) ( ∃ 𝑣 ∈ ( 𝑆 ‘ suc 𝑁 ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 ( 1st ‘ 𝑢 ) ) ∨ ∃ 𝑢 ∈ ( 𝑆 ‘ 𝑁 ) ∃ 𝑣 ∈ ( ( 𝑆 ‘ suc 𝑁 ) ∖ ( 𝑆 ‘ 𝑁 ) ) 𝑥 = ( ( 1st ‘ 𝑢 ) ⊼𝑔 ( 1st ‘ 𝑣 ) ) ) } = { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) |
233 |
33 232
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → dom { 〈 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 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) |
234 |
14 233
|
ineq12d |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( dom ( 𝑆 ‘ suc 𝑁 ) ∩ dom { 〈 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 ) ) } ) = ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) ) |
235 |
|
fmlasucdisj |
⊢ ( 𝑁 ∈ ω → ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) = ∅ ) |
236 |
235
|
ad2antrr |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( ( Fmla ‘ suc 𝑁 ) ∩ { 𝑥 ∣ ( ∃ 𝑓 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) ( ∃ 𝑔 ∈ ( Fmla ‘ suc 𝑁 ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ∨ ∃ 𝑖 ∈ ω 𝑥 = ∀𝑔 𝑖 𝑓 ) ∨ ∃ 𝑓 ∈ ( Fmla ‘ 𝑁 ) ∃ 𝑔 ∈ ( ( Fmla ‘ suc 𝑁 ) ∖ ( Fmla ‘ 𝑁 ) ) 𝑥 = ( 𝑓 ⊼𝑔 𝑔 ) ) } ) = ∅ ) |
237 |
234 236
|
eqtrd |
⊢ ( ( ( 𝑁 ∈ ω ∧ ( 𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ) ) ∧ Fun ( 𝑆 ‘ suc 𝑁 ) ) → ( dom ( 𝑆 ‘ suc 𝑁 ) ∩ dom { 〈 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 ) ) } ) = ∅ ) |