Step | Hyp | Ref
| Expression |
1 | | df-rab 3062 |
. . . . . 6
⊢ {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)} |
2 | | dfsbcq 3682 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) |
3 | 2 | cbvrexvw 3350 |
. . . . . . . . . 10
⊢
(∃𝑏 ∈
ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) |
4 | 3 | anbi2i 626 |
. . . . . . . . 9
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ ∃𝑏 ∈
ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ ∃𝑐 ∈
ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) |
5 | | r19.42v 3254 |
. . . . . . . . 9
⊢
(∃𝑐 ∈
ℕ0 (𝑎
∈ (ℕ0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ ∃𝑐 ∈
ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) |
6 | 4, 5 | bitr4i 281 |
. . . . . . . 8
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ ∃𝑏 ∈
ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) |
7 | | simpll 767 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → 𝑁 ∈
ℕ0) |
8 | | simpr 488 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → 𝑎 ∈ (ℕ0
↑m (1...𝑁))) |
9 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → 𝑐 ∈ ℕ0) |
10 | | rexrabdioph.1 |
. . . . . . . . . . . . . 14
⊢ 𝑀 = (𝑁 + 1) |
11 | 10 | mapfzcons 40130 |
. . . . . . . . . . . . 13
⊢ ((𝑁 ∈ ℕ0
∧ 𝑎 ∈
(ℕ0 ↑m (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {〈𝑀, 𝑐〉}) ∈ (ℕ0
↑m (1...𝑀))) |
12 | 7, 8, 9, 11 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → (𝑎 ∪ {〈𝑀, 𝑐〉}) ∈ (ℕ0
↑m (1...𝑀))) |
13 | 12 | adantrr 717 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ (𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {〈𝑀, 𝑐〉}) ∈ (ℕ0
↑m (1...𝑀))) |
14 | 10 | mapfzcons2 40133 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ 𝑐 ∈
ℕ0) → ((𝑎 ∪ {〈𝑀, 𝑐〉})‘𝑀) = 𝑐) |
15 | 8, 9, 14 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → ((𝑎 ∪ {〈𝑀, 𝑐〉})‘𝑀) = 𝑐) |
16 | 15 | eqcomd 2744 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → 𝑐 = ((𝑎 ∪ {〈𝑀, 𝑐〉})‘𝑀)) |
17 | 10 | mapfzcons1 40131 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑎 ∈ (ℕ0
↑m (1...𝑁))
→ ((𝑎 ∪
{〈𝑀, 𝑐〉}) ↾ (1...𝑁)) = 𝑎) |
18 | 17 | adantl 485 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → ((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁)) = 𝑎) |
19 | 18 | eqcomd 2744 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → 𝑎 = ((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁))) |
20 | 19 | sbceq1d 3685 |
. . . . . . . . . . . . . 14
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ 𝑎 ∈ (ℕ0
↑m (1...𝑁))) → ([𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁)) / 𝑢]𝜓)) |
21 | 16, 20 | sbceqbid 3687 |
. . . . . . . . . . . . 13
⊢ (((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)) |
22 | 21 | biimpd 232 |
. . . . . . . . . . . 12
⊢ (((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)) |
23 | 22 | impr 458 |
. . . . . . . . . . 11
⊢ (((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 | 19 | adantrr 717 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ0
∧ 𝑐 ∈
ℕ0) ∧ (𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁))) |
25 | | fveq1 6673 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = (𝑎 ∪ {〈𝑀, 𝑐〉}) → (𝑏‘𝑀) = ((𝑎 ∪ {〈𝑀, 𝑐〉})‘𝑀)) |
26 | | reseq1 5819 |
. . . . . . . . . . . . . . 15
⊢ (𝑏 = (𝑎 ∪ {〈𝑀, 𝑐〉}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁))) |
27 | 26 | sbceq1d 3685 |
. . . . . . . . . . . . . 14
⊢ (𝑏 = (𝑎 ∪ {〈𝑀, 𝑐〉}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁)) / 𝑢]𝜓)) |
28 | 25, 27 | sbceqbid 3687 |
. . . . . . . . . . . . 13
⊢ (𝑏 = (𝑎 ∪ {〈𝑀, 𝑐〉}) → ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {〈𝑀, 𝑐〉})‘𝑀) / 𝑣][((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁)) / 𝑢]𝜓)) |
29 | 26 | eqeq2d 2749 |
. . . . . . . . . . . . 13
⊢ (𝑏 = (𝑎 ∪ {〈𝑀, 𝑐〉}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {〈𝑀, 𝑐〉}) ↾ (1...𝑁)))) |
30 | 28, 29 | anbi12d 634 |
. . . . . . . . . . . 12
⊢ (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))))) |
31 | 30 | rspcev 3526 |
. . . . . . . . . . 11
⊢ (((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)))) |
32 | 13, 23, 24, 31 | syl12anc 836 |
. . . . . . . . . 10
⊢ (((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)))) |
33 | 32 | rexlimdva2 3197 |
. . . . . . . . 9
⊢ (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))))) |
34 | | elmapi 8459 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ (ℕ0
↑m (1...𝑀))
→ 𝑏:(1...𝑀)⟶ℕ0) |
35 | | nn0p1nn 12015 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
ℕ) |
36 | 10, 35 | eqeltrid 2837 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈ ℕ0
→ 𝑀 ∈
ℕ) |
37 | | elfz1end 13028 |
. . . . . . . . . . . . . 14
⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀)) |
38 | 36, 37 | sylib 221 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈ ℕ0
→ 𝑀 ∈ (1...𝑀)) |
39 | | ffvelrn 6859 |
. . . . . . . . . . . . 13
⊢ ((𝑏:(1...𝑀)⟶ℕ0 ∧ 𝑀 ∈ (1...𝑀)) → (𝑏‘𝑀) ∈
ℕ0) |
40 | 34, 38, 39 | syl2anr 600 |
. . . . . . . . . . . 12
⊢ ((𝑁 ∈ ℕ0
∧ 𝑏 ∈
(ℕ0 ↑m (1...𝑀))) → (𝑏‘𝑀) ∈
ℕ0) |
41 | 40 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ0
∧ 𝑏 ∈
(ℕ0 ↑m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏‘𝑀) ∈
ℕ0) |
42 | | simprr 773 |
. . . . . . . . . . . 12
⊢ (((𝑁 ∈ ℕ0
∧ 𝑏 ∈
(ℕ0 ↑m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁))) |
43 | 10 | mapfzcons1cl 40132 |
. . . . . . . . . . . . 13
⊢ (𝑏 ∈ (ℕ0
↑m (1...𝑀))
→ (𝑏 ↾
(1...𝑁)) ∈
(ℕ0 ↑m (1...𝑁))) |
44 | 43 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((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))) |
45 | 42, 44 | eqeltrd 2833 |
. . . . . . . . . . 11
⊢ (((𝑁 ∈ ℕ0
∧ 𝑏 ∈
(ℕ0 ↑m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ0
↑m (1...𝑁))) |
46 | | simprl 771 |
. . . . . . . . . . . 12
⊢ (((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) |
47 | | dfsbcq 3682 |
. . . . . . . . . . . . . 14
⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)) |
48 | 47 | sbcbidv 3736 |
. . . . . . . . . . . . 13
⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)) |
49 | 48 | ad2antll 729 |
. . . . . . . . . . . 12
⊢ (((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)) |
50 | 46, 49 | mpbird 260 |
. . . . . . . . . . 11
⊢ (((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) |
51 | | dfsbcq 3682 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑏‘𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓)) |
52 | 51 | anbi2d 632 |
. . . . . . . . . . . 12
⊢ (𝑐 = (𝑏‘𝑀) → ((𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0
↑m (1...𝑁))
∧ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓))) |
53 | 52 | rspcev 3526 |
. . . . . . . . . . 11
⊢ (((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)) |
54 | 41, 45, 50, 53 | syl12anc 836 |
. . . . . . . . . 10
⊢ (((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)) |
55 | 54 | rexlimdva2 3197 |
. . . . . . . . 9
⊢ (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))) |
56 | 33, 55 | impbid 215 |
. . . . . . . 8
⊢ (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))))) |
57 | 6, 56 | syl5bb 286 |
. . . . . . 7
⊢ (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))))) |
58 | 57 | abbidv 2802 |
. . . . . 6
⊢ (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)))}) |
59 | 1, 58 | syl5eq 2785 |
. . . . 5
⊢ (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)))}) |
60 | | nfcv 2899 |
. . . . . 6
⊢
Ⅎ𝑢(ℕ0 ↑m
(1...𝑁)) |
61 | | nfcv 2899 |
. . . . . 6
⊢
Ⅎ𝑎(ℕ0 ↑m
(1...𝑁)) |
62 | | nfv 1921 |
. . . . . 6
⊢
Ⅎ𝑎∃𝑣 ∈ ℕ0
𝜓 |
63 | | nfcv 2899 |
. . . . . . 7
⊢
Ⅎ𝑢ℕ0 |
64 | | nfcv 2899 |
. . . . . . . 8
⊢
Ⅎ𝑢𝑏 |
65 | | nfsbc1v 3700 |
. . . . . . . 8
⊢
Ⅎ𝑢[𝑎 / 𝑢]𝜓 |
66 | 64, 65 | nfsbcw 3702 |
. . . . . . 7
⊢
Ⅎ𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓 |
67 | 63, 66 | nfrex 3219 |
. . . . . 6
⊢
Ⅎ𝑢∃𝑏 ∈ ℕ0
[𝑏 / 𝑣][𝑎 / 𝑢]𝜓 |
68 | | sbceq1a 3691 |
. . . . . . . 8
⊢ (𝑢 = 𝑎 → (𝜓 ↔ [𝑎 / 𝑢]𝜓)) |
69 | 68 | rexbidv 3207 |
. . . . . . 7
⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓)) |
70 | | nfv 1921 |
. . . . . . . 8
⊢
Ⅎ𝑏[𝑎 / 𝑢]𝜓 |
71 | | nfsbc1v 3700 |
. . . . . . . 8
⊢
Ⅎ𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓 |
72 | | sbceq1a 3691 |
. . . . . . . 8
⊢ (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓 ↔ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)) |
73 | 70, 71, 72 | cbvrexw 3341 |
. . . . . . 7
⊢
(∃𝑣 ∈
ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) |
74 | 69, 73 | bitrdi 290 |
. . . . . 6
⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)) |
75 | 60, 61, 62, 67, 74 | cbvrabw 3391 |
. . . . 5
⊢ {𝑢 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑣 ∈
ℕ0 𝜓} =
{𝑎 ∈
(ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓} |
76 | | fveq1 6673 |
. . . . . . . 8
⊢ (𝑡 = 𝑏 → (𝑡‘𝑀) = (𝑏‘𝑀)) |
77 | | reseq1 5819 |
. . . . . . . . 9
⊢ (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁))) |
78 | 77 | sbceq1d 3685 |
. . . . . . . 8
⊢ (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)) |
79 | 76, 78 | sbceqbid 3687 |
. . . . . . 7
⊢ (𝑡 = 𝑏 → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)) |
80 | 79 | rexrab 3595 |
. . . . . 6
⊢
(∃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)))) |
81 | 80 | abbii 2803 |
. . . . 5
⊢ {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)))} |
82 | 59, 75, 81 | 3eqtr4g 2798 |
. . . 4
⊢ (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))}) |
83 | | fvex 6687 |
. . . . . . . 8
⊢ (𝑡‘𝑀) ∈ V |
84 | | vex 3402 |
. . . . . . . . 9
⊢ 𝑡 ∈ V |
85 | 84 | resex 5873 |
. . . . . . . 8
⊢ (𝑡 ↾ (1...𝑁)) ∈ V |
86 | | rexrabdioph.2 |
. . . . . . . . 9
⊢ (𝑣 = (𝑡‘𝑀) → (𝜓 ↔ 𝜒)) |
87 | | rexrabdioph.3 |
. . . . . . . . 9
⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒 ↔ 𝜑)) |
88 | 86, 87 | sylan9bb 513 |
. . . . . . . 8
⊢ ((𝑣 = (𝑡‘𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓 ↔ 𝜑)) |
89 | 83, 85, 88 | sbc2ie 3758 |
. . . . . . 7
⊢
([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑) |
90 | 89 | rabbii 3374 |
. . . . . 6
⊢ {𝑡 ∈ (ℕ0
↑m (1...𝑀))
∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0
↑m (1...𝑀))
∣ 𝜑} |
91 | 90 | rexeqi 3315 |
. . . . 5
⊢
(∃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 2803 |
. . . 4
⊢ {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 | 82, 92 | eqtrdi 2789 |
. . 3
⊢ (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 484 |
. 2
⊢ ((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 486 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ {𝑡 ∈
(ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈
ℕ0) |
96 | | nn0z 12086 |
. . . . . 6
⊢ (𝑁 ∈ ℕ0
→ 𝑁 ∈
ℤ) |
97 | | uzid 12339 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
(ℤ≥‘𝑁)) |
98 | | peano2uz 12383 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘𝑁) → (𝑁 + 1) ∈
(ℤ≥‘𝑁)) |
99 | 96, 97, 98 | 3syl 18 |
. . . . 5
⊢ (𝑁 ∈ ℕ0
→ (𝑁 + 1) ∈
(ℤ≥‘𝑁)) |
100 | 10, 99 | eqeltrid 2837 |
. . . 4
⊢ (𝑁 ∈ ℕ0
→ 𝑀 ∈
(ℤ≥‘𝑁)) |
101 | 100 | adantr 484 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ {𝑡 ∈
(ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ≥‘𝑁)) |
102 | | simpr 488 |
. . 3
⊢ ((𝑁 ∈ ℕ0
∧ {𝑡 ∈
(ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0
↑m (1...𝑀))
∣ 𝜑} ∈
(Dioph‘𝑀)) |
103 | | diophrex 40189 |
. . 3
⊢ ((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 1372 |
. 2
⊢ ((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 2833 |
1
⊢ ((𝑁 ∈ ℕ0
∧ {𝑡 ∈
(ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0
↑m (1...𝑁))
∣ ∃𝑣 ∈
ℕ0 𝜓} ∈
(Dioph‘𝑁)) |