Step |
Hyp |
Ref |
Expression |
1 |
|
rexrabdioph.1 |
⊢ 𝑀 = ( 𝑁 + 1 ) |
2 |
|
rexrabdioph.2 |
⊢ ( 𝑣 = ( 𝑡 ‘ 𝑀 ) → ( 𝜓 ↔ 𝜒 ) ) |
3 |
|
rexrabdioph.3 |
⊢ ( 𝑢 = ( 𝑡 ↾ ( 1 ... 𝑁 ) ) → ( 𝜒 ↔ 𝜑 ) ) |
4 |
|
df-rab |
⊢ { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 } = { 𝑎 ∣ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) } |
5 |
|
dfsbcq |
⊢ ( 𝑏 = 𝑐 → ( [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ↔ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
6 |
5
|
cbvrexvw |
⊢ ( ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ↔ ∃ 𝑐 ∈ ℕ0 [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) |
7 |
6
|
anbi2i |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ↔ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑐 ∈ ℕ0 [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
8 |
|
r19.42v |
⊢ ( ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ↔ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑐 ∈ ℕ0 [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
9 |
7 8
|
bitr4i |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ↔ ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
10 |
|
simpll |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → 𝑁 ∈ ℕ0 ) |
11 |
|
simpr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) |
12 |
|
simplr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → 𝑐 ∈ ℕ0 ) |
13 |
1
|
mapfzcons |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ 𝑐 ∈ ℕ0 ) → ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) |
14 |
10 11 12 13
|
syl3anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) |
15 |
14
|
adantrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) → ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) |
16 |
1
|
mapfzcons2 |
⊢ ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ 𝑐 ∈ ℕ0 ) → ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) = 𝑐 ) |
17 |
11 12 16
|
syl2anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) = 𝑐 ) |
18 |
17
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → 𝑐 = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) ) |
19 |
1
|
mapfzcons1 |
⊢ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) → ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) = 𝑎 ) |
20 |
19
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) = 𝑎 ) |
21 |
20
|
eqcomd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → 𝑎 = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) ) |
22 |
21
|
sbceq1d |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → ( [ 𝑎 / 𝑢 ] 𝜓 ↔ [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
23 |
18 22
|
sbceqbid |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → ( [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ↔ [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) / 𝑣 ] [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
24 |
23
|
biimpd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) → ( [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 → [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) / 𝑣 ] [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
25 |
24
|
impr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) → [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) / 𝑣 ] [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) |
26 |
21
|
adantrr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) → 𝑎 = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) ) |
27 |
|
fveq1 |
⊢ ( 𝑏 = ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) → ( 𝑏 ‘ 𝑀 ) = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) ) |
28 |
|
reseq1 |
⊢ ( 𝑏 = ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) → ( 𝑏 ↾ ( 1 ... 𝑁 ) ) = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) ) |
29 |
28
|
sbceq1d |
⊢ ( 𝑏 = ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) → ( [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ↔ [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
30 |
27 29
|
sbceqbid |
⊢ ( 𝑏 = ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) → ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ↔ [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) / 𝑣 ] [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
31 |
28
|
eqeq2d |
⊢ ( 𝑏 = ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) → ( 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ↔ 𝑎 = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) ) ) |
32 |
30 31
|
anbi12d |
⊢ ( 𝑏 = ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) → ( ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ↔ ( [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) / 𝑣 ] [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) ) ) ) |
33 |
32
|
rspcev |
⊢ ( ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∧ ( [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ‘ 𝑀 ) / 𝑣 ] [ ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( ( 𝑎 ∪ { 〈 𝑀 , 𝑐 〉 } ) ↾ ( 1 ... 𝑁 ) ) ) ) → ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) |
34 |
15 25 26 33
|
syl12anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0 ) ∧ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) → ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) |
35 |
34
|
rexlimdva2 |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) → ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) ) |
36 |
|
elmapi |
⊢ ( 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) → 𝑏 : ( 1 ... 𝑀 ) ⟶ ℕ0 ) |
37 |
|
nn0p1nn |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ ) |
38 |
1 37
|
eqeltrid |
⊢ ( 𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ ) |
39 |
|
elfz1end |
⊢ ( 𝑀 ∈ ℕ ↔ 𝑀 ∈ ( 1 ... 𝑀 ) ) |
40 |
38 39
|
sylib |
⊢ ( 𝑁 ∈ ℕ0 → 𝑀 ∈ ( 1 ... 𝑀 ) ) |
41 |
|
ffvelrn |
⊢ ( ( 𝑏 : ( 1 ... 𝑀 ) ⟶ ℕ0 ∧ 𝑀 ∈ ( 1 ... 𝑀 ) ) → ( 𝑏 ‘ 𝑀 ) ∈ ℕ0 ) |
42 |
36 40 41
|
syl2anr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) → ( 𝑏 ‘ 𝑀 ) ∈ ℕ0 ) |
43 |
42
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → ( 𝑏 ‘ 𝑀 ) ∈ ℕ0 ) |
44 |
|
simprr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) |
45 |
1
|
mapfzcons1cl |
⊢ ( 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) → ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) |
46 |
45
|
ad2antlr |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) |
47 |
44 46
|
eqeltrd |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ) |
48 |
|
simprl |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) |
49 |
|
dfsbcq |
⊢ ( 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) → ( [ 𝑎 / 𝑢 ] 𝜓 ↔ [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
50 |
49
|
sbcbidv |
⊢ ( 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) → ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ↔ [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
51 |
50
|
ad2antll |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ↔ [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
52 |
48 51
|
mpbird |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) |
53 |
|
dfsbcq |
⊢ ( 𝑐 = ( 𝑏 ‘ 𝑀 ) → ( [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ↔ [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
54 |
53
|
anbi2d |
⊢ ( 𝑐 = ( 𝑏 ‘ 𝑀 ) → ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ↔ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) ) |
55 |
54
|
rspcev |
⊢ ( ( ( 𝑏 ‘ 𝑀 ) ∈ ℕ0 ∧ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) → ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
56 |
43 47 52 55
|
syl12anc |
⊢ ( ( ( 𝑁 ∈ ℕ0 ∧ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ) ∧ ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) → ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
57 |
56
|
rexlimdva2 |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) → ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) ) |
58 |
35 57
|
impbid |
⊢ ( 𝑁 ∈ ℕ0 → ( ∃ 𝑐 ∈ ℕ0 ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ [ 𝑐 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ↔ ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) ) |
59 |
9 58
|
syl5bb |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ↔ ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) ) |
60 |
59
|
abbidv |
⊢ ( 𝑁 ∈ ℕ0 → { 𝑎 ∣ ( 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∧ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) } = { 𝑎 ∣ ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) } ) |
61 |
4 60
|
syl5eq |
⊢ ( 𝑁 ∈ ℕ0 → { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 } = { 𝑎 ∣ ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) } ) |
62 |
|
nfcv |
⊢ Ⅎ 𝑢 ( ℕ0 ↑m ( 1 ... 𝑁 ) ) |
63 |
|
nfcv |
⊢ Ⅎ 𝑎 ( ℕ0 ↑m ( 1 ... 𝑁 ) ) |
64 |
|
nfv |
⊢ Ⅎ 𝑎 ∃ 𝑣 ∈ ℕ0 𝜓 |
65 |
|
nfcv |
⊢ Ⅎ 𝑢 ℕ0 |
66 |
|
nfcv |
⊢ Ⅎ 𝑢 𝑏 |
67 |
|
nfsbc1v |
⊢ Ⅎ 𝑢 [ 𝑎 / 𝑢 ] 𝜓 |
68 |
66 67
|
nfsbcw |
⊢ Ⅎ 𝑢 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 |
69 |
65 68
|
nfrex |
⊢ Ⅎ 𝑢 ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 |
70 |
|
sbceq1a |
⊢ ( 𝑢 = 𝑎 → ( 𝜓 ↔ [ 𝑎 / 𝑢 ] 𝜓 ) ) |
71 |
70
|
rexbidv |
⊢ ( 𝑢 = 𝑎 → ( ∃ 𝑣 ∈ ℕ0 𝜓 ↔ ∃ 𝑣 ∈ ℕ0 [ 𝑎 / 𝑢 ] 𝜓 ) ) |
72 |
|
nfv |
⊢ Ⅎ 𝑏 [ 𝑎 / 𝑢 ] 𝜓 |
73 |
|
nfsbc1v |
⊢ Ⅎ 𝑣 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 |
74 |
|
sbceq1a |
⊢ ( 𝑣 = 𝑏 → ( [ 𝑎 / 𝑢 ] 𝜓 ↔ [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
75 |
72 73 74
|
cbvrexw |
⊢ ( ∃ 𝑣 ∈ ℕ0 [ 𝑎 / 𝑢 ] 𝜓 ↔ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) |
76 |
71 75
|
syl6bb |
⊢ ( 𝑢 = 𝑎 → ( ∃ 𝑣 ∈ ℕ0 𝜓 ↔ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 ) ) |
77 |
62 63 64 69 76
|
cbvrabw |
⊢ { 𝑢 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑣 ∈ ℕ0 𝜓 } = { 𝑎 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑏 ∈ ℕ0 [ 𝑏 / 𝑣 ] [ 𝑎 / 𝑢 ] 𝜓 } |
78 |
|
fveq1 |
⊢ ( 𝑡 = 𝑏 → ( 𝑡 ‘ 𝑀 ) = ( 𝑏 ‘ 𝑀 ) ) |
79 |
|
reseq1 |
⊢ ( 𝑡 = 𝑏 → ( 𝑡 ↾ ( 1 ... 𝑁 ) ) = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) |
80 |
79
|
sbceq1d |
⊢ ( 𝑡 = 𝑏 → ( [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ↔ [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
81 |
78 80
|
sbceqbid |
⊢ ( 𝑡 = 𝑏 → ( [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ↔ [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ) ) |
82 |
81
|
rexrab |
⊢ ( ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ↔ ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) ) |
83 |
82
|
abbii |
⊢ { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } = { 𝑎 ∣ ∃ 𝑏 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ( [ ( 𝑏 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑏 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ∧ 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) } |
84 |
61 77 83
|
3eqtr4g |
⊢ ( 𝑁 ∈ ℕ0 → { 𝑢 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑣 ∈ ℕ0 𝜓 } = { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } ) |
85 |
|
fvex |
⊢ ( 𝑡 ‘ 𝑀 ) ∈ V |
86 |
|
vex |
⊢ 𝑡 ∈ V |
87 |
86
|
resex |
⊢ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) ∈ V |
88 |
2 3
|
sylan9bb |
⊢ ( ( 𝑣 = ( 𝑡 ‘ 𝑀 ) ∧ 𝑢 = ( 𝑡 ↾ ( 1 ... 𝑁 ) ) ) → ( 𝜓 ↔ 𝜑 ) ) |
89 |
85 87 88
|
sbc2ie |
⊢ ( [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 ↔ 𝜑 ) |
90 |
89
|
rabbii |
⊢ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 } = { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } |
91 |
90
|
rexeqi |
⊢ ( ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ↔ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) ) |
92 |
91
|
abbii |
⊢ { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ [ ( 𝑡 ‘ 𝑀 ) / 𝑣 ] [ ( 𝑡 ↾ ( 1 ... 𝑁 ) ) / 𝑢 ] 𝜓 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } = { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } |
93 |
84 92
|
eqtrdi |
⊢ ( 𝑁 ∈ ℕ0 → { 𝑢 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑣 ∈ ℕ0 𝜓 } = { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } ) |
94 |
93
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → { 𝑢 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑣 ∈ ℕ0 𝜓 } = { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } ) |
95 |
|
simpl |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → 𝑁 ∈ ℕ0 ) |
96 |
|
nn0z |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ ) |
97 |
|
uzid |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
98 |
|
peano2uz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑁 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
99 |
96 97 98
|
3syl |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑁 ) ) |
100 |
1 99
|
eqeltrid |
⊢ ( 𝑁 ∈ ℕ0 → 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
101 |
100
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ) |
102 |
|
simpr |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) |
103 |
|
diophrex |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ 𝑀 ∈ ( ℤ≥ ‘ 𝑁 ) ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } ∈ ( Dioph ‘ 𝑁 ) ) |
104 |
95 101 102 103
|
syl3anc |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → { 𝑎 ∣ ∃ 𝑏 ∈ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } 𝑎 = ( 𝑏 ↾ ( 1 ... 𝑁 ) ) } ∈ ( Dioph ‘ 𝑁 ) ) |
105 |
94 104
|
eqeltrd |
⊢ ( ( 𝑁 ∈ ℕ0 ∧ { 𝑡 ∈ ( ℕ0 ↑m ( 1 ... 𝑀 ) ) ∣ 𝜑 } ∈ ( Dioph ‘ 𝑀 ) ) → { 𝑢 ∈ ( ℕ0 ↑m ( 1 ... 𝑁 ) ) ∣ ∃ 𝑣 ∈ ℕ0 𝜓 } ∈ ( Dioph ‘ 𝑁 ) ) |