Step | Hyp | Ref
| Expression |
1 | | eqidd 2623 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) = (𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})) |
2 | | dmeq 5324 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑅 → dom 𝑥 = dom 𝑅) |
3 | | rneq 5351 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑅 → ran 𝑥 = ran 𝑅) |
4 | 2, 3 | uneq12d 3768 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑅 → (dom 𝑥 ∪ ran 𝑥) = (dom 𝑅 ∪ ran 𝑅)) |
5 | 4 | reseq2d 5396 |
. . . . . . . . 9
⊢ (𝑥 = 𝑅 → ( I ↾ (dom 𝑥 ∪ ran 𝑥)) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) |
6 | 5 | sseq1d 3632 |
. . . . . . . 8
⊢ (𝑥 = 𝑅 → (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧)) |
7 | | id 22 |
. . . . . . . . 9
⊢ (𝑥 = 𝑅 → 𝑥 = 𝑅) |
8 | 7 | sseq1d 3632 |
. . . . . . . 8
⊢ (𝑥 = 𝑅 → (𝑥 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑧)) |
9 | 6, 8 | 3anbi12d 1400 |
. . . . . . 7
⊢ (𝑥 = 𝑅 → ((( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧))) |
10 | 9 | abbidv 2741 |
. . . . . 6
⊢ (𝑥 = 𝑅 → {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
11 | 10 | inteqd 4480 |
. . . . 5
⊢ (𝑥 = 𝑅 → ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
12 | 11 | adantl 482 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 = 𝑅) → ∩ {𝑧 ∣ (( I ↾ (dom 𝑥 ∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
13 | | drrtrcl2.2 |
. . . 4
⊢ (𝜑 → 𝑅 ∈ V) |
14 | | drrtrcl2.1 |
. . . . . . . . . 10
⊢ (𝜑 → Rel 𝑅) |
15 | | relfld 5661 |
. . . . . . . . . 10
⊢ (Rel
𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
16 | 14, 15 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) |
17 | 16 | eqcomd 2628 |
. . . . . . . 8
⊢ (𝜑 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅) |
18 | 14, 13 | rtrclreclem1 13798 |
. . . . . . . . 9
⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅)) |
19 | | id 22 |
. . . . . . . . . . 11
⊢ ((dom
𝑅 ∪ ran 𝑅) = ∪
∪ 𝑅 → (dom 𝑅 ∪ ran 𝑅) = ∪ ∪ 𝑅) |
20 | 19 | reseq2d 5396 |
. . . . . . . . . 10
⊢ ((dom
𝑅 ∪ ran 𝑅) = ∪
∪ 𝑅 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) = ( I ↾ ∪ ∪ 𝑅)) |
21 | 20 | sseq1d 3632 |
. . . . . . . . 9
⊢ ((dom
𝑅 ∪ ran 𝑅) = ∪
∪ 𝑅 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ↔ ( I ↾ ∪ ∪ 𝑅) ⊆ (t*rec‘𝑅))) |
22 | 18, 21 | syl5ibr 236 |
. . . . . . . 8
⊢ ((dom
𝑅 ∪ ran 𝑅) = ∪
∪ 𝑅 → (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))) |
23 | 17, 22 | mpcom 38 |
. . . . . . 7
⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅)) |
24 | 13 | rtrclreclem2 13799 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ⊆ (t*rec‘𝑅)) |
25 | 14, 13 | rtrclreclem3 13800 |
. . . . . . 7
⊢ (𝜑 → ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)) |
26 | | fvex 6201 |
. . . . . . . 8
⊢
(t*rec‘𝑅)
∈ V |
27 | | sseq2 3627 |
. . . . . . . . . . 11
⊢ (𝑧 = (t*rec‘𝑅) → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅))) |
28 | | sseq2 3627 |
. . . . . . . . . . 11
⊢ (𝑧 = (t*rec‘𝑅) → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ (t*rec‘𝑅))) |
29 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑧 = (t*rec‘𝑅) → 𝑧 = (t*rec‘𝑅)) |
30 | 29, 29 | coeq12d 5286 |
. . . . . . . . . . . 12
⊢ (𝑧 = (t*rec‘𝑅) → (𝑧 ∘ 𝑧) = ((t*rec‘𝑅) ∘ (t*rec‘𝑅))) |
31 | 30, 29 | sseq12d 3634 |
. . . . . . . . . . 11
⊢ (𝑧 = (t*rec‘𝑅) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))) |
32 | 27, 28, 31 | 3anbi123d 1399 |
. . . . . . . . . 10
⊢ (𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom
𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))) |
33 | 32 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → (𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))) |
34 | 33 | alrimiv 1855 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑧(𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))) |
35 | | elabgt 3347 |
. . . . . . . 8
⊢
(((t*rec‘𝑅)
∈ V ∧ ∀𝑧(𝑧 = (t*rec‘𝑅) → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅))))) → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))) |
36 | 26, 34, 35 | sylancr 695 |
. . . . . . 7
⊢ (𝜑 → ((t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*rec‘𝑅) ∧ 𝑅 ⊆ (t*rec‘𝑅) ∧ ((t*rec‘𝑅) ∘ (t*rec‘𝑅)) ⊆ (t*rec‘𝑅)))) |
37 | 23, 24, 25, 36 | mpbir3and 1245 |
. . . . . 6
⊢ (𝜑 → (t*rec‘𝑅) ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
38 | | ne0i 3921 |
. . . . . 6
⊢
((t*rec‘𝑅)
∈ {𝑧 ∣ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅) |
39 | 37, 38 | syl 17 |
. . . . 5
⊢ (𝜑 → {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅) |
40 | | intex 4820 |
. . . . 5
⊢ ({𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ≠ ∅ ↔ ∩ {𝑧
∣ (( I ↾ (dom 𝑅
∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V) |
41 | 39, 40 | sylib 208 |
. . . 4
⊢ (𝜑 → ∩ {𝑧
∣ (( I ↾ (dom 𝑅
∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ∈ V) |
42 | 1, 12, 13, 41 | fvmptd 6288 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
43 | | intss1 4492 |
. . . . 5
⊢
((t*rec‘𝑅)
∈ {𝑧 ∣ (( I
↾ (dom 𝑅 ∪ ran
𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅)) |
44 | 37, 43 | syl 17 |
. . . 4
⊢ (𝜑 → ∩ {𝑧
∣ (( I ↾ (dom 𝑅
∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ⊆ (t*rec‘𝑅)) |
45 | | vex 3203 |
. . . . . . . 8
⊢ 𝑠 ∈ V |
46 | | sseq2 3627 |
. . . . . . . . 9
⊢ (𝑧 = 𝑠 → (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ↔ ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠)) |
47 | | sseq2 3627 |
. . . . . . . . 9
⊢ (𝑧 = 𝑠 → (𝑅 ⊆ 𝑧 ↔ 𝑅 ⊆ 𝑠)) |
48 | | id 22 |
. . . . . . . . . . 11
⊢ (𝑧 = 𝑠 → 𝑧 = 𝑠) |
49 | 48, 48 | coeq12d 5286 |
. . . . . . . . . 10
⊢ (𝑧 = 𝑠 → (𝑧 ∘ 𝑧) = (𝑠 ∘ 𝑠)) |
50 | 49, 48 | sseq12d 3634 |
. . . . . . . . 9
⊢ (𝑧 = 𝑠 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑠 ∘ 𝑠) ⊆ 𝑠)) |
51 | 46, 47, 50 | 3anbi123d 1399 |
. . . . . . . 8
⊢ (𝑧 = 𝑠 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧) ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠))) |
52 | 45, 51 | elab 3350 |
. . . . . . 7
⊢ (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠)) |
53 | 14, 13 | rtrclreclem4 13801 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑠((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) |
54 | 53 | 19.21bi 2059 |
. . . . . . 7
⊢ (𝜑 → ((( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑠 ∧ 𝑅 ⊆ 𝑠 ∧ (𝑠 ∘ 𝑠) ⊆ 𝑠) → (t*rec‘𝑅) ⊆ 𝑠)) |
55 | 52, 54 | syl5bi 232 |
. . . . . 6
⊢ (𝜑 → (𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} → (t*rec‘𝑅) ⊆ 𝑠)) |
56 | 55 | ralrimiv 2965 |
. . . . 5
⊢ (𝜑 → ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠) |
57 | | ssint 4493 |
. . . . 5
⊢
((t*rec‘𝑅)
⊆ ∩ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} ↔ ∀𝑠 ∈ {𝑧 ∣ (( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} (t*rec‘𝑅) ⊆ 𝑠) |
58 | 56, 57 | sylibr 224 |
. . . 4
⊢ (𝜑 → (t*rec‘𝑅) ⊆ ∩ {𝑧
∣ (( I ↾ (dom 𝑅
∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
59 | 44, 58 | eqssd 3620 |
. . 3
⊢ (𝜑 → ∩ {𝑧
∣ (( I ↾ (dom 𝑅
∪ ran 𝑅)) ⊆ 𝑧 ∧ 𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)} = (t*rec‘𝑅)) |
60 | 42, 59 | eqtrd 2656 |
. 2
⊢ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)) |
61 | | df-rtrcl 13727 |
. . 3
⊢ t* =
(𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) |
62 | | fveq1 6190 |
. . . . 5
⊢ (t* =
(𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → (t*‘𝑅) = ((𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅)) |
63 | 62 | eqeq1d 2624 |
. . . 4
⊢ (t* =
(𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → ((t*‘𝑅) = (t*rec‘𝑅) ↔ ((𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))) |
64 | 63 | imbi2d 330 |
. . 3
⊢ (t* =
(𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)}) → ((𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) ↔ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅)))) |
65 | 61, 64 | ax-mp 5 |
. 2
⊢ ((𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) ↔ (𝜑 → ((𝑥 ∈ V ↦ ∩ {𝑧
∣ (( I ↾ (dom 𝑥
∪ ran 𝑥)) ⊆ 𝑧 ∧ 𝑥 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})‘𝑅) = (t*rec‘𝑅))) |
66 | 60, 65 | mpbir 221 |
1
⊢ (𝜑 → (t*‘𝑅) = (t*rec‘𝑅)) |