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Mirrors > Home > MPE Home > Th. List > xkotopon | Structured version Visualization version GIF version |
Description: The base set of the compact-open topology. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
xkouni.1 | ⊢ 𝐽 = (𝑆 ↑ko 𝑅) |
Ref | Expression |
---|---|
xkotopon | ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xkouni.1 | . . 3 ⊢ 𝐽 = (𝑆 ↑ko 𝑅) | |
2 | xkotop 22124 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top) | |
3 | 1, 2 | eqeltrid 2914 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ Top) |
4 | 1 | xkouni 22135 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽) |
5 | istopon 21448 | . 2 ⊢ (𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)) ↔ (𝐽 ∈ Top ∧ (𝑅 Cn 𝑆) = ∪ 𝐽)) | |
6 | 3, 4, 5 | sylanbrc 583 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∪ cuni 4830 ‘cfv 6348 (class class class)co 7145 Topctop 21429 TopOnctopon 21446 Cn ccn 21760 ↑ko cxko 22097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-fin 8501 df-fi 8863 df-rest 16684 df-topgen 16705 df-top 21430 df-topon 21447 df-bases 21482 df-cmp 21923 df-xko 22099 |
This theorem is referenced by: xkoccn 22155 xkopjcn 22192 xkoco1cn 22193 xkoco2cn 22194 xkococn 22196 cnmptkp 22216 cnmptk1 22217 cnmpt1k 22218 cnmptkk 22219 xkofvcn 22220 cnmptk1p 22221 cnmptk2 22222 xkoinjcn 22223 xkocnv 22350 xkohmeo 22351 symgtgp 22637 efmndtmd 43997 |
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