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Mirrors > Home > MPE Home > Th. List > kgencmp | Structured version Visualization version GIF version |
Description: The compact generator topology is the same as the original topology on compact subspaces. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
kgencmp | ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kgenftop 22151 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ Top) | |
2 | 1 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝑘Gen‘𝐽) ∈ Top) |
3 | kgenss 22154 | . . . 4 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
5 | ssrest 21787 | . . 3 ⊢ (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐽 ⊆ (𝑘Gen‘𝐽)) → (𝐽 ↾t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾)) | |
6 | 2, 4, 5 | syl2anc 586 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ⊆ ((𝑘Gen‘𝐽) ↾t 𝐾)) |
7 | cmptop 22006 | . . . . . 6 ⊢ ((𝐽 ↾t 𝐾) ∈ Comp → (𝐽 ↾t 𝐾) ∈ Top) | |
8 | 7 | adantl 484 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) ∈ Top) |
9 | restrcl 21768 | . . . . . 6 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → (𝐽 ∈ V ∧ 𝐾 ∈ V)) | |
10 | 9 | simprd 498 | . . . . 5 ⊢ ((𝐽 ↾t 𝐾) ∈ Top → 𝐾 ∈ V) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → 𝐾 ∈ V) |
12 | restval 16703 | . . . 4 ⊢ (((𝑘Gen‘𝐽) ∈ Top ∧ 𝐾 ∈ V) → ((𝑘Gen‘𝐽) ↾t 𝐾) = ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥 ∩ 𝐾))) | |
13 | 2, 11, 12 | syl2anc 586 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) = ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥 ∩ 𝐾))) |
14 | simpr 487 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → 𝑥 ∈ (𝑘Gen‘𝐽)) | |
15 | simplr 767 | . . . . . 6 ⊢ (((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝐽 ↾t 𝐾) ∈ Comp) | |
16 | kgeni 22148 | . . . . . 6 ⊢ ((𝑥 ∈ (𝑘Gen‘𝐽) ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝑥 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾)) | |
17 | 14, 15, 16 | syl2anc 586 | . . . . 5 ⊢ (((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) ∧ 𝑥 ∈ (𝑘Gen‘𝐽)) → (𝑥 ∩ 𝐾) ∈ (𝐽 ↾t 𝐾)) |
18 | 17 | fmpttd 6882 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥 ∩ 𝐾)):(𝑘Gen‘𝐽)⟶(𝐽 ↾t 𝐾)) |
19 | 18 | frnd 6524 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ran (𝑥 ∈ (𝑘Gen‘𝐽) ↦ (𝑥 ∩ 𝐾)) ⊆ (𝐽 ↾t 𝐾)) |
20 | 13, 19 | eqsstrd 4008 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → ((𝑘Gen‘𝐽) ↾t 𝐾) ⊆ (𝐽 ↾t 𝐾)) |
21 | 6, 20 | eqssd 3987 | 1 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ↾t 𝐾) ∈ Comp) → (𝐽 ↾t 𝐾) = ((𝑘Gen‘𝐽) ↾t 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∩ cin 3938 ⊆ wss 3939 ↦ cmpt 5149 ran crn 5559 ‘cfv 6358 (class class class)co 7159 ↾t crest 16697 Topctop 21504 Compccmp 21997 𝑘Genckgen 22144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-oadd 8109 df-er 8292 df-en 8513 df-fin 8516 df-fi 8878 df-rest 16699 df-topgen 16720 df-top 21505 df-topon 21522 df-bases 21557 df-cmp 21998 df-kgen 22145 |
This theorem is referenced by: kgencmp2 22157 kgenidm 22158 |
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