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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 22193 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top) | |
3 | 1, 2 | eqeltrid 2894 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ Top) |
4 | 1 | xkouni 22204 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽) |
5 | istopon 21517 | . 2 ⊢ (𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)) ↔ (𝐽 ∈ Top ∧ (𝑅 Cn 𝑆) = ∪ 𝐽)) | |
6 | 3, 4, 5 | sylanbrc 586 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∪ cuni 4800 ‘cfv 6324 (class class class)co 7135 Topctop 21498 TopOnctopon 21515 Cn ccn 21829 ↑ko cxko 22166 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-fin 8496 df-fi 8859 df-rest 16688 df-topgen 16709 df-top 21499 df-topon 21516 df-bases 21551 df-cmp 21992 df-xko 22168 |
This theorem is referenced by: xkoccn 22224 xkopjcn 22261 xkoco1cn 22262 xkoco2cn 22263 xkococn 22265 cnmptkp 22285 cnmptk1 22286 cnmpt1k 22287 cnmptkk 22288 xkofvcn 22289 cnmptk1p 22290 cnmptk2 22291 xkoinjcn 22292 xkocnv 22419 xkohmeo 22420 efmndtmd 22706 symgtgp 22711 |
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