Step | Hyp | Ref
| Expression |
1 | | tfrcl.x |
. . 3
⊢ (𝜑 → Ord 𝑋) |
2 | | ordelon 4366 |
. . 3
⊢ ((Ord
𝑋 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ On) |
3 | 1, 2 | sylan 281 |
. 2
⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → 𝐶 ∈ On) |
4 | | eleq1 2233 |
. . . . 5
⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑋 ↔ 𝑤 ∈ 𝑋)) |
5 | 4 | anbi2d 461 |
. . . 4
⊢ (𝑧 = 𝑤 → ((𝜑 ∧ 𝑧 ∈ 𝑋) ↔ (𝜑 ∧ 𝑤 ∈ 𝑋))) |
6 | | feq2 5329 |
. . . . . 6
⊢ (𝑧 = 𝑤 → (𝑔:𝑧⟶𝑆 ↔ 𝑔:𝑤⟶𝑆)) |
7 | | raleq 2665 |
. . . . . 6
⊢ (𝑧 = 𝑤 → (∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
8 | 6, 7 | anbi12d 470 |
. . . . 5
⊢ (𝑧 = 𝑤 → ((𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
9 | 8 | exbidv 1818 |
. . . 4
⊢ (𝑧 = 𝑤 → (∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
10 | 5, 9 | imbi12d 233 |
. . 3
⊢ (𝑧 = 𝑤 → (((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
11 | | eleq1 2233 |
. . . . 5
⊢ (𝑧 = 𝐶 → (𝑧 ∈ 𝑋 ↔ 𝐶 ∈ 𝑋)) |
12 | 11 | anbi2d 461 |
. . . 4
⊢ (𝑧 = 𝐶 → ((𝜑 ∧ 𝑧 ∈ 𝑋) ↔ (𝜑 ∧ 𝐶 ∈ 𝑋))) |
13 | | feq2 5329 |
. . . . . 6
⊢ (𝑧 = 𝐶 → (𝑔:𝑧⟶𝑆 ↔ 𝑔:𝐶⟶𝑆)) |
14 | | raleq 2665 |
. . . . . 6
⊢ (𝑧 = 𝐶 → (∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
15 | 13, 14 | anbi12d 470 |
. . . . 5
⊢ (𝑧 = 𝐶 → ((𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
16 | 15 | exbidv 1818 |
. . . 4
⊢ (𝑧 = 𝐶 → (∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
17 | 12, 16 | imbi12d 233 |
. . 3
⊢ (𝑧 = 𝐶 → (((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
18 | | tfrcl.f |
. . . . . . . . 9
⊢ 𝐹 = recs(𝐺) |
19 | | tfrcl.g |
. . . . . . . . . 10
⊢ (𝜑 → Fun 𝐺) |
20 | 19 | ad3antrrr 489 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → Fun 𝐺) |
21 | 1 | ad3antrrr 489 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → Ord 𝑋) |
22 | | tfrcl.ex |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑓:𝑥⟶𝑆) → (𝐺‘𝑓) ∈ 𝑆) |
23 | 22 | 3expia 1200 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)) |
24 | 23 | alrimiv 1867 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆)) |
25 | | feq1 5328 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = ℎ → (𝑓:𝑥⟶𝑆 ↔ ℎ:𝑥⟶𝑆)) |
26 | | fveq2 5494 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = ℎ → (𝐺‘𝑓) = (𝐺‘ℎ)) |
27 | 26 | eleq1d 2239 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = ℎ → ((𝐺‘𝑓) ∈ 𝑆 ↔ (𝐺‘ℎ) ∈ 𝑆)) |
28 | 25, 27 | imbi12d 233 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = ℎ → ((𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ (ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆))) |
29 | 28 | cbvalv 1910 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑓(𝑓:𝑥⟶𝑆 → (𝐺‘𝑓) ∈ 𝑆) ↔ ∀ℎ(ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆)) |
30 | 24, 29 | sylib 121 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀ℎ(ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆)) |
31 | 30 | 19.21bi 1551 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℎ:𝑥⟶𝑆 → (𝐺‘ℎ) ∈ 𝑆)) |
32 | 31 | 3impia 1195 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆) |
33 | 32 | 3adant1r 1226 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆) |
34 | 33 | 3adant1r 1226 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆) |
35 | 34 | 3adant1r 1226 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋 ∧ ℎ:𝑥⟶𝑆) → (𝐺‘ℎ) ∈ 𝑆) |
36 | | tfrcllemsucfn.1 |
. . . . . . . . . 10
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} |
37 | | fveq1 5493 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = ℎ → (𝑓‘𝑦) = (ℎ‘𝑦)) |
38 | | reseq1 4883 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = ℎ → (𝑓 ↾ 𝑦) = (ℎ ↾ 𝑦)) |
39 | 38 | fveq2d 5498 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = ℎ → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘(ℎ ↾ 𝑦))) |
40 | 37, 39 | eqeq12d 2185 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = ℎ → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))) |
41 | 40 | ralbidv 2470 |
. . . . . . . . . . . . 13
⊢ (𝑓 = ℎ → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))) |
42 | 25, 41 | anbi12d 470 |
. . . . . . . . . . . 12
⊢ (𝑓 = ℎ → ((𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦))))) |
43 | 42 | rexbidv 2471 |
. . . . . . . . . . 11
⊢ (𝑓 = ℎ → (∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ 𝑋 (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦))))) |
44 | 43 | cbvabv 2295 |
. . . . . . . . . 10
⊢ {𝑓 ∣ ∃𝑥 ∈ 𝑋 (𝑓:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {ℎ ∣ ∃𝑥 ∈ 𝑋 (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))} |
45 | 36, 44 | eqtri 2191 |
. . . . . . . . 9
⊢ 𝐴 = {ℎ ∣ ∃𝑥 ∈ 𝑋 (ℎ:𝑥⟶𝑆 ∧ ∀𝑦 ∈ 𝑥 (ℎ‘𝑦) = (𝐺‘(ℎ ↾ 𝑦)))} |
46 | | feq1 5328 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = 𝑎 → (𝑟:𝑡⟶𝑆 ↔ 𝑎:𝑡⟶𝑆)) |
47 | | eleq1 2233 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = 𝑎 → (𝑟 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴)) |
48 | | id 19 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 𝑎 → 𝑟 = 𝑎) |
49 | | fveq2 5494 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = 𝑎 → (𝐺‘𝑟) = (𝐺‘𝑎)) |
50 | 49 | opeq2d 3770 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = 𝑎 → 〈𝑡, (𝐺‘𝑟)〉 = 〈𝑡, (𝐺‘𝑎)〉) |
51 | 50 | sneqd 3594 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = 𝑎 → {〈𝑡, (𝐺‘𝑟)〉} = {〈𝑡, (𝐺‘𝑎)〉}) |
52 | 48, 51 | uneq12d 3282 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = 𝑎 → (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉}) = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉})) |
53 | 52 | eqeq2d 2182 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = 𝑎 → (𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉}) ↔ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉}))) |
54 | 46, 47, 53 | 3anbi123d 1307 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = 𝑎 → ((𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉})) ↔ (𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉})))) |
55 | 54 | cbvexv 1911 |
. . . . . . . . . . . . 13
⊢
(∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉})) ↔ ∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉}))) |
56 | 55 | rexbii 2477 |
. . . . . . . . . . . 12
⊢
(∃𝑡 ∈
𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉})) ↔ ∃𝑡 ∈ 𝑧 ∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉}))) |
57 | | feq2 5329 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑏 → (𝑎:𝑡⟶𝑆 ↔ 𝑎:𝑏⟶𝑆)) |
58 | | opeq1 3763 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 = 𝑏 → 〈𝑡, (𝐺‘𝑎)〉 = 〈𝑏, (𝐺‘𝑎)〉) |
59 | 58 | sneqd 3594 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑏 → {〈𝑡, (𝐺‘𝑎)〉} = {〈𝑏, (𝐺‘𝑎)〉}) |
60 | 59 | uneq2d 3281 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑏 → (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉}) = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})) |
61 | 60 | eqeq2d 2182 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑏 → (𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉}) ↔ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))) |
62 | 57, 61 | 3anbi13d 1309 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑏 → ((𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉})) ↔ (𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})))) |
63 | 62 | exbidv 1818 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑏 → (∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉})) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})))) |
64 | 63 | cbvrexv 2697 |
. . . . . . . . . . . 12
⊢
(∃𝑡 ∈
𝑧 ∃𝑎(𝑎:𝑡⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑡, (𝐺‘𝑎)〉})) ↔ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))) |
65 | 56, 64 | bitri 183 |
. . . . . . . . . . 11
⊢
(∃𝑡 ∈
𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉})) ↔ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))) |
66 | 65 | abbii 2286 |
. . . . . . . . . 10
⊢ {𝑠 ∣ ∃𝑡 ∈ 𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉}))} = {𝑠 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))} |
67 | | eqeq1 2177 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑑 → (𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}) ↔ 𝑑 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))) |
68 | 67 | 3anbi3d 1313 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑑 → ((𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})) ↔ (𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})))) |
69 | 68 | exbidv 1818 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑑 → (∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})))) |
70 | 69 | rexbidv 2471 |
. . . . . . . . . . 11
⊢ (𝑠 = 𝑑 → (∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})) ↔ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉})))) |
71 | 70 | cbvabv 2295 |
. . . . . . . . . 10
⊢ {𝑠 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑠 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))} = {𝑑 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))} |
72 | 66, 71 | eqtri 2191 |
. . . . . . . . 9
⊢ {𝑠 ∣ ∃𝑡 ∈ 𝑧 ∃𝑟(𝑟:𝑡⟶𝑆 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 = (𝑟 ∪ {〈𝑡, (𝐺‘𝑟)〉}))} = {𝑑 ∣ ∃𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ 𝑎 ∈ 𝐴 ∧ 𝑑 = (𝑎 ∪ {〈𝑏, (𝐺‘𝑎)〉}))} |
73 | | tfrcllemaccex.u |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
74 | 73 | adantlr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
75 | 74 | adantlr 474 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
76 | 75 | adantlr 474 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑥 ∈ ∪ 𝑋) → suc 𝑥 ∈ 𝑋) |
77 | | simpr 109 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋) |
78 | | simpr 109 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → 𝑏 ∈ 𝑧) |
79 | | simplr 525 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → 𝑧 ∈ 𝑋) |
80 | | ordtr1 4371 |
. . . . . . . . . . . . . 14
⊢ (Ord
𝑋 → ((𝑏 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) → 𝑏 ∈ 𝑋)) |
81 | 1, 80 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑏 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) → 𝑏 ∈ 𝑋)) |
82 | 81 | ad4antr 491 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → ((𝑏 ∈ 𝑧 ∧ 𝑧 ∈ 𝑋) → 𝑏 ∈ 𝑋)) |
83 | 78, 79, 82 | mp2and 431 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → 𝑏 ∈ 𝑋) |
84 | | eleq1 2233 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑏 → (𝑤 ∈ 𝑋 ↔ 𝑏 ∈ 𝑋)) |
85 | | feq2 5329 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑏 → (𝑔:𝑤⟶𝑆 ↔ 𝑔:𝑏⟶𝑆)) |
86 | | raleq 2665 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 = 𝑏 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
87 | 85, 86 | anbi12d 470 |
. . . . . . . . . . . . . . 15
⊢ (𝑤 = 𝑏 → ((𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
88 | 87 | exbidv 1818 |
. . . . . . . . . . . . . 14
⊢ (𝑤 = 𝑏 → (∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
89 | 84, 88 | imbi12d 233 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑏 → ((𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝑏 ∈ 𝑋 → ∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
90 | | simpllr 529 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
91 | 89, 90, 78 | rspcdva 2839 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → (𝑏 ∈ 𝑋 → ∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
92 | | feq1 5328 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑎 → (𝑔:𝑏⟶𝑆 ↔ 𝑎:𝑏⟶𝑆)) |
93 | | fveq1 5493 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑔 = 𝑎 → (𝑔‘𝑢) = (𝑎‘𝑢)) |
94 | | reseq1 4883 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑔 = 𝑎 → (𝑔 ↾ 𝑢) = (𝑎 ↾ 𝑢)) |
95 | 94 | fveq2d 5498 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑔 = 𝑎 → (𝐺‘(𝑔 ↾ 𝑢)) = (𝐺‘(𝑎 ↾ 𝑢))) |
96 | 93, 95 | eqeq12d 2185 |
. . . . . . . . . . . . . . . 16
⊢ (𝑔 = 𝑎 → ((𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)))) |
97 | 96 | ralbidv 2470 |
. . . . . . . . . . . . . . 15
⊢ (𝑔 = 𝑎 → (∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)))) |
98 | 92, 97 | anbi12d 470 |
. . . . . . . . . . . . . 14
⊢ (𝑔 = 𝑎 → ((𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ (𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))))) |
99 | 98 | cbvexv 1911 |
. . . . . . . . . . . . 13
⊢
(∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)))) |
100 | | fveq2 5494 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑐 → (𝑎‘𝑢) = (𝑎‘𝑐)) |
101 | | reseq2 4884 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑢 = 𝑐 → (𝑎 ↾ 𝑢) = (𝑎 ↾ 𝑐)) |
102 | 101 | fveq2d 5498 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = 𝑐 → (𝐺‘(𝑎 ↾ 𝑢)) = (𝐺‘(𝑎 ↾ 𝑐))) |
103 | 100, 102 | eqeq12d 2185 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 = 𝑐 → ((𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)) ↔ (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
104 | 103 | cbvralv 2696 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑢 ∈
𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢)) ↔ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))) |
105 | 104 | anbi2i 454 |
. . . . . . . . . . . . . 14
⊢ ((𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))) ↔ (𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
106 | 105 | exbii 1598 |
. . . . . . . . . . . . 13
⊢
(∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑎‘𝑢) = (𝐺‘(𝑎 ↾ 𝑢))) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
107 | 99, 106 | bitri 183 |
. . . . . . . . . . . 12
⊢
(∃𝑔(𝑔:𝑏⟶𝑆 ∧ ∀𝑢 ∈ 𝑏 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))) ↔ ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
108 | 91, 107 | syl6ib 160 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → (𝑏 ∈ 𝑋 → ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐))))) |
109 | 83, 108 | mpd 13 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑧) → ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
110 | 109 | ralrimiva 2543 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → ∀𝑏 ∈ 𝑧 ∃𝑎(𝑎:𝑏⟶𝑆 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))) |
111 | 18, 20, 21, 35, 45, 72, 76, 77, 110 | tfrcllemex 6337 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → ∃ℎ(ℎ:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢)))) |
112 | | feq1 5328 |
. . . . . . . . . 10
⊢ (ℎ = 𝑔 → (ℎ:𝑧⟶𝑆 ↔ 𝑔:𝑧⟶𝑆)) |
113 | | fveq1 5493 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑔 → (ℎ‘𝑢) = (𝑔‘𝑢)) |
114 | | reseq1 4883 |
. . . . . . . . . . . . 13
⊢ (ℎ = 𝑔 → (ℎ ↾ 𝑢) = (𝑔 ↾ 𝑢)) |
115 | 114 | fveq2d 5498 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑔 → (𝐺‘(ℎ ↾ 𝑢)) = (𝐺‘(𝑔 ↾ 𝑢))) |
116 | 113, 115 | eqeq12d 2185 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑔 → ((ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢)) ↔ (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
117 | 116 | ralbidv 2470 |
. . . . . . . . . 10
⊢ (ℎ = 𝑔 → (∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
118 | 112, 117 | anbi12d 470 |
. . . . . . . . 9
⊢ (ℎ = 𝑔 → ((ℎ:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢))) ↔ (𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
119 | 118 | cbvexv 1911 |
. . . . . . . 8
⊢
(∃ℎ(ℎ:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (ℎ‘𝑢) = (𝐺‘(ℎ ↾ 𝑢))) ↔ ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
120 | 111, 119 | sylib 121 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |
121 | 120 | exp31 362 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ On) → (∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
122 | 121 | expcom 115 |
. . . . 5
⊢ (𝑧 ∈ On → (𝜑 → (∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))) |
123 | 122 | a2d 26 |
. . . 4
⊢ (𝑧 ∈ On → ((𝜑 → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) → (𝜑 → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))))) |
124 | | impexp 261 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝜑 → (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
125 | 124 | ralbii 2476 |
. . . . 5
⊢
(∀𝑤 ∈
𝑧 ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ ∀𝑤 ∈ 𝑧 (𝜑 → (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
126 | | r19.21v 2547 |
. . . . 5
⊢
(∀𝑤 ∈
𝑧 (𝜑 → (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) ↔ (𝜑 → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
127 | 125, 126 | bitri 183 |
. . . 4
⊢
(∀𝑤 ∈
𝑧 ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝜑 → ∀𝑤 ∈ 𝑧 (𝑤 ∈ 𝑋 → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
128 | | impexp 261 |
. . . 4
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) ↔ (𝜑 → (𝑧 ∈ 𝑋 → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
129 | 123, 127,
128 | 3imtr4g 204 |
. . 3
⊢ (𝑧 ∈ On → (∀𝑤 ∈ 𝑧 ((𝜑 ∧ 𝑤 ∈ 𝑋) → ∃𝑔(𝑔:𝑤⟶𝑆 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) → ((𝜑 ∧ 𝑧 ∈ 𝑋) → ∃𝑔(𝑔:𝑧⟶𝑆 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))))) |
130 | 10, 17, 129 | tfis3 4568 |
. 2
⊢ (𝐶 ∈ On → ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢))))) |
131 | 3, 130 | mpcom 36 |
1
⊢ ((𝜑 ∧ 𝐶 ∈ 𝑋) → ∃𝑔(𝑔:𝐶⟶𝑆 ∧ ∀𝑢 ∈ 𝐶 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))) |