Step | Hyp | Ref
| Expression |
1 | | kgentopon 21341 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) →
(𝑘Gen‘𝐽)
∈ (TopOn‘𝑋)) |
2 | | iscn 21039 |
. . 3
⊢
(((𝑘Gen‘𝐽) ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽)))) |
3 | 1, 2 | sylan 488 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽)))) |
4 | | elkgen 21339 |
. . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → ((◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝐹 “ 𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
5 | 4 | ad2antrr 762 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝐹 “ 𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
6 | | cnvimass 5485 |
. . . . . . . 8
⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹 |
7 | | fdm 6051 |
. . . . . . . . 9
⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋) |
8 | 7 | adantl 482 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → dom 𝐹 = 𝑋) |
9 | 6, 8 | syl5sseq 3653 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (◡𝐹 “ 𝑥) ⊆ 𝑋) |
10 | 9 | biantrurd 529 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ((◡𝐹 “ 𝑥) ⊆ 𝑋 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))))) |
11 | 5, 10 | bitr4d 271 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
12 | 11 | ralbidv 2986 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑥 ∈ 𝐾 ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
13 | | simpll 790 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐽 ∈ (TopOn‘𝑋)) |
14 | | elpwi 4168 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝒫 𝑋 → 𝑘 ⊆ 𝑋) |
15 | | resttopon 20965 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑘 ⊆ 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘)) |
16 | 13, 14, 15 | syl2an 494 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘)) |
17 | | simpllr 799 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → 𝐾 ∈ (TopOn‘𝑌)) |
18 | | iscn 21039 |
. . . . . . . . . . 11
⊢ (((𝐽 ↾t 𝑘) ∈ (TopOn‘𝑘) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ((𝐹 ↾ 𝑘):𝑘⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘)))) |
19 | 16, 17, 18 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ((𝐹 ↾ 𝑘):𝑘⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘)))) |
20 | | simpr 477 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝐹:𝑋⟶𝑌) |
21 | | fssres 6070 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑘 ⊆ 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌) |
22 | 20, 14, 21 | syl2an 494 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (𝐹 ↾ 𝑘):𝑘⟶𝑌) |
23 | 22 | biantrurd 529 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ((𝐹 ↾ 𝑘):𝑘⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘)))) |
24 | 19, 23 | bitr4d 271 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘))) |
25 | | cnvresima 5623 |
. . . . . . . . . . 11
⊢ (◡(𝐹 ↾ 𝑘) “ 𝑥) = ((◡𝐹 “ 𝑥) ∩ 𝑘) |
26 | 25 | eleq1i 2692 |
. . . . . . . . . 10
⊢ ((◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) |
27 | 26 | ralbii 2980 |
. . . . . . . . 9
⊢
(∀𝑥 ∈
𝐾 (◡(𝐹 ↾ 𝑘) “ 𝑥) ∈ (𝐽 ↾t 𝑘) ↔ ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) |
28 | 24, 27 | syl6bb 276 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → ((𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾) ↔ ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) |
29 | 28 | imbi2d 330 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
30 | | r19.21v 2960 |
. . . . . . 7
⊢
(∀𝑥 ∈
𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ((𝐽 ↾t 𝑘) ∈ Comp → ∀𝑥 ∈ 𝐾 ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) |
31 | 29, 30 | syl6bbr 278 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑘 ∈ 𝒫 𝑋) → (((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
32 | 31 | ralbidva 2985 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)) ↔ ∀𝑘 ∈ 𝒫 𝑋∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)))) |
33 | | ralcom 3098 |
. . . . 5
⊢
(∀𝑥 ∈
𝐾 ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋∀𝑥 ∈ 𝐾 ((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘))) |
34 | 32, 33 | syl6rbbr 279 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝐾 ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → ((◡𝐹 “ 𝑥) ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))) |
35 | 12, 34 | bitrd 268 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾)))) |
36 | 35 | pm5.32da 673 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝐾 (◡𝐹 “ 𝑥) ∈ (𝑘Gen‘𝐽)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))) |
37 | 3, 36 | bitrd 268 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ ((𝑘Gen‘𝐽) Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑘 ∈ 𝒫 𝑋((𝐽 ↾t 𝑘) ∈ Comp → (𝐹 ↾ 𝑘) ∈ ((𝐽 ↾t 𝑘) Cn 𝐾))))) |