Step | Hyp | Ref
| Expression |
1 | | imadmres 5872 |
. . . . 5
⊢ (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ (𝐴 ∖ 𝑈)) |
2 | | dmres 5659 |
. . . . . . 7
⊢ dom
(𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ dom 𝐹) |
3 | | kqval.2 |
. . . . . . . . . . 11
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) |
4 | 3 | kqffn 21906 |
. . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋) |
5 | 4 | adantr 474 |
. . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → 𝐹 Fn 𝑋) |
6 | | fndm 6227 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) |
7 | 5, 6 | syl 17 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom 𝐹 = 𝑋) |
8 | 7 | ineq2d 4043 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐴 ∖ 𝑈) ∩ dom 𝐹) = ((𝐴 ∖ 𝑈) ∩ 𝑋)) |
9 | 2, 8 | syl5eq 2873 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → dom (𝐹 ↾ (𝐴 ∖ 𝑈)) = ((𝐴 ∖ 𝑈) ∩ 𝑋)) |
10 | 9 | imaeq2d 5711 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ dom (𝐹 ↾ (𝐴 ∖ 𝑈))) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋))) |
11 | 1, 10 | syl5eqr 2875 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) = (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋))) |
12 | | indif1 4103 |
. . . . . 6
⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) = ((𝐴 ∩ 𝑋) ∖ 𝑈) |
13 | | inss2 4060 |
. . . . . . 7
⊢ (𝐴 ∩ 𝑋) ⊆ 𝑋 |
14 | | ssdif 3974 |
. . . . . . 7
⊢ ((𝐴 ∩ 𝑋) ⊆ 𝑋 → ((𝐴 ∩ 𝑋) ∖ 𝑈) ⊆ (𝑋 ∖ 𝑈)) |
15 | 13, 14 | ax-mp 5 |
. . . . . 6
⊢ ((𝐴 ∩ 𝑋) ∖ 𝑈) ⊆ (𝑋 ∖ 𝑈) |
16 | 12, 15 | eqsstri 3860 |
. . . . 5
⊢ ((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈) |
17 | | imass2 5746 |
. . . . 5
⊢ (((𝐴 ∖ 𝑈) ∩ 𝑋) ⊆ (𝑋 ∖ 𝑈) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈))) |
18 | 16, 17 | mp1i 13 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ ((𝐴 ∖ 𝑈) ∩ 𝑋)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈))) |
19 | 11, 18 | eqsstrd 3864 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈))) |
20 | | sslin 4065 |
. . 3
⊢ ((𝐹 “ (𝐴 ∖ 𝑈)) ⊆ (𝐹 “ (𝑋 ∖ 𝑈)) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈)))) |
21 | 19, 20 | syl 17 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈)))) |
22 | | eldifn 3962 |
. . . . . . 7
⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → ¬ 𝑤 ∈ 𝑈) |
23 | 22 | adantl 475 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ 𝑤 ∈ 𝑈) |
24 | | simpll 783 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝐽 ∈ (TopOn‘𝑋)) |
25 | | simplr 785 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑈 ∈ 𝐽) |
26 | | eldifi 3961 |
. . . . . . . 8
⊢ (𝑤 ∈ (𝑋 ∖ 𝑈) → 𝑤 ∈ 𝑋) |
27 | 26 | adantl 475 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → 𝑤 ∈ 𝑋) |
28 | 3 | kqfvima 21911 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝑤 ∈ 𝑋) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))) |
29 | 24, 25, 27, 28 | syl3anc 1494 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → (𝑤 ∈ 𝑈 ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))) |
30 | 23, 29 | mtbid 316 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) ∧ 𝑤 ∈ (𝑋 ∖ 𝑈)) → ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)) |
31 | 30 | ralrimiva 3175 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈)) |
32 | | difss 3966 |
. . . . 5
⊢ (𝑋 ∖ 𝑈) ⊆ 𝑋 |
33 | | eleq1 2894 |
. . . . . . 7
⊢ (𝑧 = (𝐹‘𝑤) → (𝑧 ∈ (𝐹 “ 𝑈) ↔ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))) |
34 | 33 | notbid 310 |
. . . . . 6
⊢ (𝑧 = (𝐹‘𝑤) → (¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))) |
35 | 34 | ralima 6759 |
. . . . 5
⊢ ((𝐹 Fn 𝑋 ∧ (𝑋 ∖ 𝑈) ⊆ 𝑋) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))) |
36 | 5, 32, 35 | sylancl 580 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → (∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑈) ¬ (𝐹‘𝑤) ∈ (𝐹 “ 𝑈))) |
37 | 31, 36 | mpbird 249 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈)) |
38 | | disjr 4244 |
. . 3
⊢ (((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅ ↔ ∀𝑧 ∈ (𝐹 “ (𝑋 ∖ 𝑈)) ¬ 𝑧 ∈ (𝐹 “ 𝑈)) |
39 | 37, 38 | sylibr 226 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅) |
40 | | sseq0 4202 |
. 2
⊢ ((((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) ⊆ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) ∧ ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝑋 ∖ 𝑈))) = ∅) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅) |
41 | 21, 39, 40 | syl2anc 579 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽) → ((𝐹 “ 𝑈) ∩ (𝐹 “ (𝐴 ∖ 𝑈))) = ∅) |