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 30274 |
. . 3
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖))) |
7 | | simp1 1054 |
. . . . . 6
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝜒) |
8 | | simp2 1055 |
. . . . . 6
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑖 ∈ 𝑛) |
9 | | bnj1128.7 |
. . . . . . . . 9
⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) |
10 | | nfv 1830 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑖 ∈ 𝑛 |
11 | | nfra1 2925 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃) |
12 | 9, 11 | nfxfr 1771 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜏 |
13 | | nfv 1830 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜒 |
14 | 10, 12, 13 | nf3an 1819 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) |
15 | | nfv 1830 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑓‘𝑖) ⊆ 𝐴 |
16 | 14, 15 | nfim 1813 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
17 | 16 | nf5ri 2053 |
. . . . . . . . . . . 12
⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) → ∀𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)) |
18 | 3 | bnj1098 30108 |
. . . . . . . . . . . . . . . . 17
⊢
∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) |
19 | | simpl 472 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ≠ ∅) |
20 | | simpr1 1060 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ∈ 𝑛) |
21 | 5 | bnj1232 30128 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → 𝑛 ∈ 𝐷) |
22 | 21 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷) |
23 | 22 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑛 ∈ 𝐷) |
24 | 19, 20, 23 | 3jca 1235 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) |
25 | 18, 24 | bnj1101 30109 |
. . . . . . . . . . . . . . . 16
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) |
26 | | ancl 567 |
. . . . . . . . . . . . . . . 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 30043 |
. . . . . . . . . . . . . . 15
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
28 | | df-3an 1033 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) ↔ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
29 | 28 | imbi2i 325 |
. . . . . . . . . . . . . . . 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 1764 |
. . . . . . . . . . . . . . 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 220 |
. . . . . . . . . . . . . 14
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
32 | | bnj213 30206 |
. . . . . . . . . . . . . . . 16
⊢
pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
33 | 32 | bnj226 30056 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
34 | | simp21 1087 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 ∈ 𝑛) |
35 | | simp3r 1083 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗) |
36 | | biid 250 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷) |
37 | | biid 250 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛) |
38 | | bnj1128.8 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑) |
39 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝑗 ∈ V |
40 | | sbcg 3470 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑 ↔ 𝜑)) |
41 | 39, 40 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
([𝑗 / 𝑖]𝜑 ↔ 𝜑) |
42 | 38, 41 | bitr2i 264 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 ↔ 𝜑′) |
43 | | bnj1128.9 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
44 | 2, 43 | bnj1039 30293 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
45 | 2, 44 | bitr4i 266 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜓 ↔ 𝜓′) |
46 | 36, 37, 42, 45 | bnj887 30089 |
. . . . . . . . . . . . . . . . . . . . 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 30294 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
49 | 46, 5, 48 | 3bitr4i 291 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 ↔ 𝜒′) |
50 | 48 | bnj1254 30134 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒′ → 𝜓′) |
51 | 49, 50 | sylbi 206 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → 𝜓′) |
52 | 51 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜓′) |
53 | 52 | 3ad2ant2 1076 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝜓′) |
54 | | simp3l 1082 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ 𝑛) |
55 | 22 | 3ad2ant2 1076 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑛 ∈ 𝐷) |
56 | 3 | bnj923 30092 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
57 | | elnn 6967 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑗 ∈ ω) |
58 | 56, 57 | sylan2 490 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑗 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → 𝑗 ∈ ω) |
59 | 54, 55, 58 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ ω) |
60 | 44 | bnj589 30233 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
61 | | rsp 2913 |
. . . . . . . . . . . . . . . . . . 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 206 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜓′ → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))) |
63 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛)) |
64 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = suc 𝑗 → (𝑓‘𝑖) = (𝑓‘suc 𝑗)) |
65 | 64 | eqeq1d 2612 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = suc 𝑗 → ((𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ↔ (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
66 | 63, 65 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . 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)))) |
67 | 66 | imbi2d 329 |
. . . . . . . . . . . . . . . . . 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))))) |
68 | 62, 67 | syl5ibr 235 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = suc 𝑗 → (𝜓′ → (𝑗 ∈ ω → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))))) |
69 | 35, 53, 59, 68 | syl3c 64 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑖 ∈ 𝑛 → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
70 | 34, 69 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)) |
71 | 33, 70 | bnj1262 30135 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) ⊆ 𝐴) |
72 | 31, 71 | bnj1023 30105 |
. . . . . . . . . . . . 13
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴) |
73 | 5 | bnj1247 30133 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝜑) |
74 | 73 | 3ad2ant3 1077 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜑) |
75 | | bnj213 30206 |
. . . . . . . . . . . . . . 15
⊢
pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴 |
76 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅)) |
77 | 1 | biimpi 205 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
78 | 76, 77 | sylan9eq 2664 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) |
79 | 75, 78 | bnj1262 30135 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) ⊆ 𝐴) |
80 | 74, 79 | sylan2 490 |
. . . . . . . . . . . . 13
⊢ ((𝑖 = ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴) |
81 | 72, 80 | bnj1109 30111 |
. . . . . . . . . . . 12
⊢
∃𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
82 | 17, 81 | bnj1131 30112 |
. . . . . . . . . . 11
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
83 | 82 | 3expia 1259 |
. . . . . . . . . 10
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
84 | | bnj1128.6 |
. . . . . . . . . 10
⊢ (𝜃 ↔ (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
85 | 83, 84 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) |
86 | 3, 5, 9, 85 | bnj1133 30311 |
. . . . . . . 8
⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃) |
87 | 84 | ralbii 2963 |
. . . . . . . 8
⊢
(∀𝑖 ∈
𝑛 𝜃 ↔ ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
88 | 86, 87 | sylib 207 |
. . . . . . 7
⊢ (𝜒 → ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
89 | | rsp 2913 |
. . . . . . 7
⊢
(∀𝑖 ∈
𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴) → (𝑖 ∈ 𝑛 → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))) |
90 | 88, 89 | syl 17 |
. . . . . 6
⊢ (𝜒 → (𝑖 ∈ 𝑛 → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴))) |
91 | 7, 8, 7, 90 | syl3c 64 |
. . . . 5
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → (𝑓‘𝑖) ⊆ 𝐴) |
92 | | simp3 1056 |
. . . . 5
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ (𝑓‘𝑖)) |
93 | 91, 92 | sseldd 3569 |
. . . 4
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ 𝐴) |
94 | 93 | 2eximi 1753 |
. . 3
⊢
(∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → ∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
95 | 6, 94 | bnj593 30069 |
. 2
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
96 | | 19.9v 1883 |
. . 3
⊢
(∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
97 | | 19.9v 1883 |
. . 3
⊢
(∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑖 𝑌 ∈ 𝐴) |
98 | | 19.9v 1883 |
. . 3
⊢
(∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) |
99 | 96, 97, 98 | 3bitri 285 |
. 2
⊢
(∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) |
100 | 95, 99 | sylib 207 |
1
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴) |