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 34964 |
. . 3
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖))) |
| 7 | | simp1 1137 |
. . . . . 6
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝜒) |
| 8 | | simp2 1138 |
. . . . . 6
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑖 ∈ 𝑛) |
| 9 | | bnj1128.7 |
. . . . . . . . 9
⊢ (𝜏 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃)) |
| 10 | | nfv 1914 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗 𝑖 ∈ 𝑛 |
| 11 | | nfra1 3284 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑗∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜃) |
| 12 | 9, 11 | nfxfr 1853 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜏 |
| 13 | | nfv 1914 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑗𝜒 |
| 14 | 10, 12, 13 | nf3an 1901 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) |
| 15 | | nfv 1914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑗(𝑓‘𝑖) ⊆ 𝐴 |
| 16 | 14, 15 | nfim 1896 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
| 17 | 16 | nf5ri 2195 |
. . . . . . . . . . . 12
⊢ (((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) → ∀𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴)) |
| 18 | 3 | bnj1098 34797 |
. . . . . . . . . . . . . . . . 17
⊢
∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) |
| 19 | | simpl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ≠ ∅) |
| 20 | | simpr1 1195 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑖 ∈ 𝑛) |
| 21 | 5 | bnj1232 34817 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 → 𝑛 ∈ 𝐷) |
| 22 | 21 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷) |
| 23 | 22 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → 𝑛 ∈ 𝐷) |
| 24 | 19, 20, 23 | 3jca 1129 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)) |
| 25 | 18, 24 | bnj1101 34798 |
. . . . . . . . . . . . . . . 16
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) |
| 26 | | ancl 544 |
. . . . . . . . . . . . . . . 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 34737 |
. . . . . . . . . . . . . . 15
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
| 28 | | df-3an 1089 |
. . . . . . . . . . . . . . . . 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 1848 |
. . . . . . . . . . . . . . 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 231 |
. . . . . . . . . . . . . 14
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))) |
| 32 | | bnj213 34896 |
. . . . . . . . . . . . . . . 16
⊢
pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
| 33 | 32 | bnj226 34748 |
. . . . . . . . . . . . . . 15
⊢ ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 |
| 34 | | simp21 1207 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 ∈ 𝑛) |
| 35 | | simp3r 1203 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑖 = suc 𝑗) |
| 36 | | biid 261 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑛 ∈ 𝐷 ↔ 𝑛 ∈ 𝐷) |
| 37 | | biid 261 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓 Fn 𝑛 ↔ 𝑓 Fn 𝑛) |
| 38 | | bnj1128.8 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑′ ↔ [𝑗 / 𝑖]𝜑) |
| 39 | | vex 3484 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ 𝑗 ∈ V |
| 40 | | sbcg 3863 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑗 ∈ V → ([𝑗 / 𝑖]𝜑 ↔ 𝜑)) |
| 41 | 39, 40 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
([𝑗 / 𝑖]𝜑 ↔ 𝜑) |
| 42 | 38, 41 | bitr2i 276 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 ↔ 𝜑′) |
| 43 | | bnj1128.9 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜓′ ↔ [𝑗 / 𝑖]𝜓) |
| 44 | 2, 43 | bnj1039 34985 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
| 45 | 2, 44 | bitr4i 278 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜓 ↔ 𝜓′) |
| 46 | 36, 37, 42, 45 | bnj887 34779 |
. . . . . . . . . . . . . . . . . . . . 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 34986 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜒′ ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑′ ∧ 𝜓′)) |
| 49 | 46, 5, 48 | 3bitr4i 303 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒 ↔ 𝜒′) |
| 50 | 48 | bnj1254 34823 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜒′ → 𝜓′) |
| 51 | 49, 50 | sylbi 217 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜒 → 𝜓′) |
| 52 | 51 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜓′) |
| 53 | 52 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝜓′) |
| 54 | | simp3l 1202 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑗 ∈ 𝑛) |
| 55 | 22 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → 𝑛 ∈ 𝐷) |
| 56 | 3 | bnj923 34782 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω) |
| 57 | | elnn 7898 |
. . . . . . . . . . . . . . . . . . 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 34923 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜓′ ↔ ∀𝑗 ∈ ω (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅))) |
| 61 | | rsp 3247 |
. . . . . . . . . . . . . . . . . . 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 217 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜓′ → (𝑗 ∈ ω → (suc 𝑗 ∈ 𝑛 → (𝑓‘suc 𝑗) = ∪ 𝑦 ∈ (𝑓‘𝑗) pred(𝑦, 𝐴, 𝑅)))) |
| 63 | | eleq1 2829 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = suc 𝑗 → (𝑖 ∈ 𝑛 ↔ suc 𝑗 ∈ 𝑛)) |
| 64 | | fveqeq2 6915 |
. . . . . . . . . . . . . . . . . . . 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 344 |
. . . . . . . . . . . . . . . . . . 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 340 |
. . . . . . . . . . . . . . . . . 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 | imbitrrid 246 |
. . . . . . . . . . . . . . . . 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 34824 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗)) → (𝑓‘𝑖) ⊆ 𝐴) |
| 71 | 31, 70 | bnj1023 34794 |
. . . . . . . . . . . . 13
⊢
∃𝑗((𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴) |
| 72 | 5 | bnj1247 34822 |
. . . . . . . . . . . . . . 15
⊢ (𝜒 → 𝜑) |
| 73 | 72 | 3ad2ant3 1136 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → 𝜑) |
| 74 | | bnj213 34896 |
. . . . . . . . . . . . . . 15
⊢
pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴 |
| 75 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅)) |
| 76 | 1 | biimpi 216 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
| 77 | 75, 76 | sylan9eq 2797 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) |
| 78 | 74, 77 | bnj1262 34824 |
. . . . . . . . . . . . . 14
⊢ ((𝑖 = ∅ ∧ 𝜑) → (𝑓‘𝑖) ⊆ 𝐴) |
| 79 | 73, 78 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝑖 = ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒)) → (𝑓‘𝑖) ⊆ 𝐴) |
| 80 | 71, 79 | bnj1109 34800 |
. . . . . . . . . . . 12
⊢
∃𝑗((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
| 81 | 17, 80 | bnj1131 34801 |
. . . . . . . . . . 11
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏 ∧ 𝜒) → (𝑓‘𝑖) ⊆ 𝐴) |
| 82 | 81 | 3expia 1122 |
. . . . . . . . . 10
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
| 83 | | bnj1128.6 |
. . . . . . . . . 10
⊢ (𝜃 ↔ (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
| 84 | 82, 83 | sylibr 234 |
. . . . . . . . 9
⊢ ((𝑖 ∈ 𝑛 ∧ 𝜏) → 𝜃) |
| 85 | 3, 5, 9, 84 | bnj1133 35003 |
. . . . . . . 8
⊢ (𝜒 → ∀𝑖 ∈ 𝑛 𝜃) |
| 86 | 83 | ralbii 3093 |
. . . . . . . 8
⊢
(∀𝑖 ∈
𝑛 𝜃 ↔ ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
| 87 | 85, 86 | sylib 218 |
. . . . . . 7
⊢ (𝜒 → ∀𝑖 ∈ 𝑛 (𝜒 → (𝑓‘𝑖) ⊆ 𝐴)) |
| 88 | | rsp 3247 |
. . . . . . 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 1139 |
. . . . 5
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ (𝑓‘𝑖)) |
| 92 | 90, 91 | sseldd 3984 |
. . . 4
⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → 𝑌 ∈ 𝐴) |
| 93 | 92 | 2eximi 1836 |
. . 3
⊢
(∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑌 ∈ (𝑓‘𝑖)) → ∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
| 94 | 6, 93 | bnj593 34759 |
. 2
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → ∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
| 95 | | 19.9v 1983 |
. . 3
⊢
(∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑛∃𝑖 𝑌 ∈ 𝐴) |
| 96 | | 19.9v 1983 |
. . 3
⊢
(∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ ∃𝑖 𝑌 ∈ 𝐴) |
| 97 | | 19.9v 1983 |
. . 3
⊢
(∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) |
| 98 | 95, 96, 97 | 3bitri 297 |
. 2
⊢
(∃𝑓∃𝑛∃𝑖 𝑌 ∈ 𝐴 ↔ 𝑌 ∈ 𝐴) |
| 99 | 94, 98 | sylib 218 |
1
⊢ (𝑌 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑌 ∈ 𝐴) |