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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 23752 | . . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) | |
| 2 | kgenidm 23757 | . . . 4 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽) | |
| 3 | eqimss 3996 | . . . 4 ⊢ ((𝑘Gen‘𝐽) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽) | |
| 4 | 2, 3 | syl 18 | . . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽) |
| 5 | 1, 4 | jca 521 | . 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
| 6 | simpr 490 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ⊆ 𝐽) | |
| 7 | kgenss 23753 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
| 8 | 7 | adantr 486 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
| 9 | 6, 8 | eqssd 3955 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) = 𝐽) |
| 10 | kgenf 23751 | . . . . . 6 ⊢ 𝑘Gen:Top⟶Top | |
| 11 | ffn 6709 | . . . . . 6 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top) | |
| 12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ 𝑘Gen Fn Top |
| 13 | fnfvelrn 7079 | . . . . 5 ⊢ ((𝑘Gen Fn Top ∧ 𝐽 ∈ Top) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) | |
| 14 | 12, 13 | mpan 703 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) |
| 15 | 14 | adantr 486 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) |
| 16 | 9, 15 | eqeltrrd 2866 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ∈ ran 𝑘Gen) |
| 17 | 5, 16 | impbii 212 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⊆ wss 3906 ran crn 5664 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 Topctop 23102 𝑘Genckgen 23743 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-en 8950 df-fin 8953 df-fi 9378 df-rest 17499 df-topgen 17520 df-top 23103 df-topon 23120 df-bases 23155 df-cmp 23596 df-kgen 23744 |
| This theorem is used by: iskgen3 23759 llycmpkgen2 23760 1stckgen 23764 txkgen 23862 qtopkgen 23920 |
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