Proof of Theorem bnj1128
Step | Hyp | Ref
| Expression |
1 | | bnj1128.1 |
. . . 4
⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
2 | | bnj1128.2 |
. . . 4
⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
3 | | bnj1128.3 |
. . . 4
⊢ 𝐷 = (ω ∖
{∅}) |
4 | | bnj1128.4 |
. . . 4
⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)} |
5 | | bnj1128.5 |
. . . 4
⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
6 | 1, 2, 3, 4, 5 | bnj981 32927 |
. . 3
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖))) |
7 | | simp1 1135 |
. . . . . 6
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝜒) |
8 | | simp2 1136 |
. . . . . 6
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑖 ∈ 𝑛) |
9 | | bnj1128.7 |
. . . . . . . . 9
⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) |
10 | | nfv 1917 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑖 ∈ 𝑛 |
11 | | nfra1 3144 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃) |
12 | 9, 11 | nfxfr 1855 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜏 |
13 | | nfv 1917 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜒 |
14 | 10, 12, 13 | nf3an 1904 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) |
15 | | nfv 1917 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑓‘𝑖) ⊆ 𝐴 |
16 | 14, 15 | nfim 1899 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
17 | 16 | nf5ri 2188 |
. . . . . . . . . . . 12
⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) → ∀𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)) |
18 | 3 | bnj1098 32760 |
. . . . . . . . . . . . . . . . 17
⊢
∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) |
19 | | simpl 483 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ≠ ∅) |
20 | | simpr1 1193 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ∈ 𝑛) |
21 | 5 | bnj1232 32780 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → 𝑛 ∈ 𝐷) |
22 | 21 | 3ad2ant3 1134 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷) |
23 | 22 | adantl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑛 ∈ 𝐷) |
24 | 19, 20, 23 | 3jca 1127 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) |
25 | 18, 24 | bnj1101 32761 |
. . . . . . . . . . . . . . . 16
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) |
26 | | ancl 545 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))) |
27 | 25, 26 | bnj101 32699 |
. . . . . . . . . . . . . . 15
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
28 | | df-3an 1088 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
29 | 28 | imbi2i 336 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) ↔ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))) |
30 | 29 | exbii 1850 |
. . . . . . . . . . . . . . 15
⊢
(∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)))) |
31 | 27, 30 | mpbir 230 |
. . . . . . . . . . . . . 14
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
32 | | bnj213 32859 |
. . . . . . . . . . . . . . . 16
⊢
pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
33 | 32 | bnj226 32710 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
34 | | simp21 1205 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 ∈ 𝑛) |
35 | | simp3r 1201 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗) |
36 | | biid 260 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷) |
37 | | biid 260 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛) |
38 | | bnj1128.8 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑) |
39 | | vex 3435 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝑗 ∈ V |
40 | | sbcg 3796 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑 ↔ 𝜑)) |
41 | 39, 40 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
([𝑗 / 𝑖]𝜑 ↔ 𝜑) |
42 | 38, 41 | bitr2i 275 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 ↔ 𝜑′) |
43 | | bnj1128.9 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
44 | 2, 43 | bnj1039 32948 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
45 | 2, 44 | bitr4i 277 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜓 ↔ 𝜓′) |
46 | 36, 37, 42, 45 | bnj887 32742 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
47 | | bnj1128.10 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜒′ ↔ [𝑗 / 𝑖]𝜒) |
48 | 38, 43, 5, 47 | bnj1040 32949 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
49 | 46, 5, 48 | 3bitr4i 303 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 ↔ 𝜒′) |
50 | 48 | bnj1254 32786 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒′ → 𝜓′) |
51 | 49, 50 | sylbi 216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → 𝜓′) |
52 | 51 | 3ad2ant3 1134 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜓′) |
53 | 52 | 3ad2ant2 1133 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝜓′) |
54 | | simp3l 1200 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ 𝑛) |
55 | 22 | 3ad2ant2 1133 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑛 ∈ 𝐷) |
56 | 3 | bnj923 32745 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
57 | | elnn 7723 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑗 ∈ ω) |
58 | 56, 57 | sylan2 593 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → 𝑗 ∈ ω) |
59 | 54, 55, 58 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ ω) |
60 | 44 | bnj589 32886 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
61 | | rsp 3131 |
. . . . . . . . . . . . . . . . . . 19
⊢
(∀𝑗 ∈
ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))) |
62 | 60, 61 | sylbi 216 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜓′ → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))) |
63 | | eleq1 2826 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛)) |
64 | | fveqeq2 6785 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = suc 𝑗 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
65 | 63, 64 | imbi12d 345 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = suc 𝑗 → ((𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) ↔ (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))) |
66 | 65 | imbi2d 341 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = suc 𝑗 → ((𝑗 ∈ ω → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) ↔ (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))) |
67 | 62, 66 | syl5ibr 245 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = suc 𝑗 → (𝜓′ → (𝑗 ∈ ω → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))) |
68 | 35, 53, 59, 67 | syl3c 66 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
69 | 34, 68 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) |
70 | 33, 69 | bnj1262 32787 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) ⊆ 𝐴) |
71 | 31, 70 | bnj1023 32757 |
. . . . . . . . . . . . 13
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴) |
72 | 5 | bnj1247 32785 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝜑) |
73 | 72 | 3ad2ant3 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜑) |
74 | | bnj213 32859 |
. . . . . . . . . . . . . . 15
⊢
pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴 |
75 | | fveq2 6776 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅)) |
76 | 1 | biimpi 215 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
77 | 75, 76 | sylan9eq 2798 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) |
78 | 74, 77 | bnj1262 32787 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) ⊆ 𝐴) |
79 | 73, 78 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑖 = ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴) |
80 | 71, 79 | bnj1109 32763 |
. . . . . . . . . . . 12
⊢
∃𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
81 | 17, 80 | bnj1131 32764 |
. . . . . . . . . . 11
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
82 | 81 | 3expia 1120 |
. . . . . . . . . 10
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
83 | | bnj1128.6 |
. . . . . . . . . 10
⊢ (𝜃 ↔ (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
84 | 82, 83 | sylibr 233 |
. . . . . . . . 9
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) |
85 | 3, 5, 9, 84 | bnj1133 32966 |
. . . . . . . 8
⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃) |
86 | 83 | ralbii 3092 |
. . . . . . . 8
⊢
(∀𝑖 ∈
𝑛 𝜃 ↔ ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
87 | 85, 86 | sylib 217 |
. . . . . . 7
⊢ (𝜒 → ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
88 | | rsp 3131 |
. . . . . . 7
⊢
(∀𝑖 ∈
𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴) → (𝑖 ∈ 𝑛 → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))) |
89 | 87, 88 | syl 17 |
. . . . . 6
⊢ (𝜒 → (𝑖 ∈ 𝑛 → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))) |
90 | 7, 8, 7, 89 | syl3c 66 |
. . . . 5
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → (𝑓‘𝑖) ⊆ 𝐴) |
91 | | simp3 1137 |
. . . . 5
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ (𝑓‘𝑖)) |
92 | 90, 91 | sseldd 3923 |
. . . 4
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ 𝐴) |
93 | 92 | 2eximi 1838 |
. . 3
⊢
(∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → ∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
94 | 6, 93 | bnj593 32722 |
. 2
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
95 | | 19.9v 1987 |
. . 3
⊢
(∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
96 | | 19.9v 1987 |
. . 3
⊢
(∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑖 𝑌 ∈ 𝐴) |
97 | | 19.9v 1987 |
. . 3
⊢
(∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) |
98 | 95, 96, 97 | 3bitri 297 |
. 2
⊢
(∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) |
99 | 94, 98 | sylib 217 |
1
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴) |