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Mirrors > Home > MPE Home > Th. List > iskgen2 | Structured version Visualization version GIF version |
Description: A space is compactly generated iff it contains its image under the compact generator. (Contributed by Mario Carneiro, 20-Mar-2015.) |
Ref | Expression |
---|---|
iskgen2 | ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kgentop 22153 | . . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) | |
2 | kgenidm 22158 | . . . 4 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽) | |
3 | eqimss 4026 | . . . 4 ⊢ ((𝑘Gen‘𝐽) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽) |
5 | 1, 4 | jca 514 | . 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
6 | simpr 487 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ⊆ 𝐽) | |
7 | kgenss 22154 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
9 | 6, 8 | eqssd 3987 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) = 𝐽) |
10 | kgenf 22152 | . . . . . 6 ⊢ 𝑘Gen:Top⟶Top | |
11 | ffn 6517 | . . . . . 6 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ 𝑘Gen Fn Top |
13 | fnfvelrn 6851 | . . . . 5 ⊢ ((𝑘Gen Fn Top ∧ 𝐽 ∈ Top) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) | |
14 | 12, 13 | mpan 688 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) |
15 | 14 | adantr 483 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) |
16 | 9, 15 | eqeltrrd 2917 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ∈ ran 𝑘Gen) |
17 | 5, 16 | impbii 211 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ran crn 5559 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 Topctop 21504 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: iskgen3 22160 llycmpkgen2 22161 1stckgen 22165 txkgen 22263 qtopkgen 22321 |
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