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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 22891 | . . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → 𝐽 ∈ Top) | |
2 | kgenidm 22896 | . . . 4 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) = 𝐽) | |
3 | eqimss 4000 | . . . 4 ⊢ ((𝑘Gen‘𝐽) = 𝐽 → (𝑘Gen‘𝐽) ⊆ 𝐽) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝑘Gen‘𝐽) ⊆ 𝐽) |
5 | 1, 4 | jca 512 | . 2 ⊢ (𝐽 ∈ ran 𝑘Gen → (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
6 | simpr 485 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ⊆ 𝐽) | |
7 | kgenss 22892 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (𝑘Gen‘𝐽)) | |
8 | 7 | adantr 481 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ⊆ (𝑘Gen‘𝐽)) |
9 | 6, 8 | eqssd 3961 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) = 𝐽) |
10 | kgenf 22890 | . . . . . 6 ⊢ 𝑘Gen:Top⟶Top | |
11 | ffn 6668 | . . . . . 6 ⊢ (𝑘Gen:Top⟶Top → 𝑘Gen Fn Top) | |
12 | 10, 11 | ax-mp 5 | . . . . 5 ⊢ 𝑘Gen Fn Top |
13 | fnfvelrn 7031 | . . . . 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 481 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → (𝑘Gen‘𝐽) ∈ ran 𝑘Gen) |
16 | 9, 15 | eqeltrrd 2839 | . 2 ⊢ ((𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽) → 𝐽 ∈ ran 𝑘Gen) |
17 | 5, 16 | impbii 208 | 1 ⊢ (𝐽 ∈ ran 𝑘Gen ↔ (𝐽 ∈ Top ∧ (𝑘Gen‘𝐽) ⊆ 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3910 ran crn 5634 Fn wfn 6491 ⟶wf 6492 ‘cfv 6496 Topctop 22240 𝑘Genckgen 22882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-en 8883 df-fin 8886 df-fi 9346 df-rest 17303 df-topgen 17324 df-top 22241 df-topon 22258 df-bases 22294 df-cmp 22736 df-kgen 22883 |
This theorem is referenced by: iskgen3 22898 llycmpkgen2 22899 1stckgen 22903 txkgen 23001 qtopkgen 23059 |
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