Step | Hyp | Ref
| Expression |
1 | | simp-6l 809 |
. . . . . . . 8
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋)) |
2 | | simp-7l 813 |
. . . . . . . . . 10
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑈 ∈ (UnifOn‘𝑋)) |
3 | | simp-4r 803 |
. . . . . . . . . 10
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑤 ∈ 𝑈) |
4 | | simplr 785 |
. . . . . . . . . 10
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑢 ∈ 𝑈) |
5 | | ustincl 22388 |
. . . . . . . . . 10
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑤 ∈ 𝑈 ∧ 𝑢 ∈ 𝑈) → (𝑤 ∩ 𝑢) ∈ 𝑈) |
6 | 2, 3, 4, 5 | syl3anc 1494 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑤 ∩ 𝑢) ∈ 𝑈) |
7 | | simpllr 793 |
. . . . . . . . . 10
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → 𝑎 = (𝑤 “ {𝑝})) |
8 | | ineq12 4038 |
. . . . . . . . . . 11
⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))) |
9 | | vex 3417 |
. . . . . . . . . . . 12
⊢ 𝑝 ∈ V |
10 | | inimasn 5795 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ V → ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝}))) |
11 | 9, 10 | ax-mp 5 |
. . . . . . . . . . 11
⊢ ((𝑤 ∩ 𝑢) “ {𝑝}) = ((𝑤 “ {𝑝}) ∩ (𝑢 “ {𝑝})) |
12 | 8, 11 | syl6eqr 2879 |
. . . . . . . . . 10
⊢ ((𝑎 = (𝑤 “ {𝑝}) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝})) |
13 | 7, 12 | sylancom 582 |
. . . . . . . . 9
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝})) |
14 | | imaeq1 5706 |
. . . . . . . . . 10
⊢ (𝑥 = (𝑤 ∩ 𝑢) → (𝑥 “ {𝑝}) = ((𝑤 ∩ 𝑢) “ {𝑝})) |
15 | 14 | rspceeqv 3544 |
. . . . . . . . 9
⊢ (((𝑤 ∩ 𝑢) ∈ 𝑈 ∧ (𝑎 ∩ 𝑏) = ((𝑤 ∩ 𝑢) “ {𝑝})) → ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})) |
16 | 6, 13, 15 | syl2anc 579 |
. . . . . . . 8
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})) |
17 | | vex 3417 |
. . . . . . . . . . 11
⊢ 𝑎 ∈ V |
18 | 17 | inex1 5026 |
. . . . . . . . . 10
⊢ (𝑎 ∩ 𝑏) ∈ V |
19 | | utopustuq.1 |
. . . . . . . . . . 11
⊢ 𝑁 = (𝑝 ∈ 𝑋 ↦ ran (𝑣 ∈ 𝑈 ↦ (𝑣 “ {𝑝}))) |
20 | 19 | ustuqtoplem 22420 |
. . . . . . . . . 10
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ (𝑎 ∩ 𝑏) ∈ V) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝}))) |
21 | 18, 20 | mpan2 682 |
. . . . . . . . 9
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ((𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝}))) |
22 | 21 | biimpar 471 |
. . . . . . . 8
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ ∃𝑥 ∈ 𝑈 (𝑎 ∩ 𝑏) = (𝑥 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝)) |
23 | 1, 16, 22 | syl2anc 579 |
. . . . . . 7
⊢
((((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) ∧ 𝑢 ∈ 𝑈) ∧ 𝑏 = (𝑢 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝)) |
24 | | simp-4l 801 |
. . . . . . . 8
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋)) |
25 | | simpllr 793 |
. . . . . . . 8
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → 𝑏 ∈ (𝑁‘𝑝)) |
26 | 19 | ustuqtoplem 22420 |
. . . . . . . . . 10
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ V) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))) |
27 | 26 | elvd 3419 |
. . . . . . . . 9
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑏 ∈ (𝑁‘𝑝) ↔ ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝}))) |
28 | 27 | biimpa 470 |
. . . . . . . 8
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑏 ∈ (𝑁‘𝑝)) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})) |
29 | 24, 25, 28 | syl2anc 579 |
. . . . . . 7
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → ∃𝑢 ∈ 𝑈 𝑏 = (𝑢 “ {𝑝})) |
30 | 23, 29 | r19.29a 3288 |
. . . . . 6
⊢
((((((𝑈 ∈
(UnifOn‘𝑋) ∧
𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) ∧ 𝑤 ∈ 𝑈) ∧ 𝑎 = (𝑤 “ {𝑝})) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝)) |
31 | 19 | ustuqtoplem 22420 |
. . . . . . . . 9
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ V) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))) |
32 | 31 | elvd 3419 |
. . . . . . . 8
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (𝑎 ∈ (𝑁‘𝑝) ↔ ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝}))) |
33 | 32 | biimpa 470 |
. . . . . . 7
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})) |
34 | 33 | adantr 474 |
. . . . . 6
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → ∃𝑤 ∈ 𝑈 𝑎 = (𝑤 “ {𝑝})) |
35 | 30, 34 | r19.29a 3288 |
. . . . 5
⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) ∧ 𝑏 ∈ (𝑁‘𝑝)) → (𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝)) |
36 | 35 | ralrimiva 3175 |
. . . 4
⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) ∧ 𝑎 ∈ (𝑁‘𝑝)) → ∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝)) |
37 | 36 | ralrimiva 3175 |
. . 3
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → ∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝)) |
38 | | fvex 6450 |
. . . 4
⊢ (𝑁‘𝑝) ∈ V |
39 | | inficl 8606 |
. . . 4
⊢ ((𝑁‘𝑝) ∈ V → (∀𝑎 ∈ (𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝))) |
40 | 38, 39 | ax-mp 5 |
. . 3
⊢
(∀𝑎 ∈
(𝑁‘𝑝)∀𝑏 ∈ (𝑁‘𝑝)(𝑎 ∩ 𝑏) ∈ (𝑁‘𝑝) ↔ (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝)) |
41 | 37, 40 | sylib 210 |
. 2
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) = (𝑁‘𝑝)) |
42 | | eqimss 3882 |
. 2
⊢
((fi‘(𝑁‘𝑝)) = (𝑁‘𝑝) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝)) |
43 | 41, 42 | syl 17 |
1
⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑝 ∈ 𝑋) → (fi‘(𝑁‘𝑝)) ⊆ (𝑁‘𝑝)) |