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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 22196 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) ∈ Top) | |
3 | 1, 2 | eqeltrid 2917 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ Top) |
4 | 1 | xkouni 22207 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 Cn 𝑆) = ∪ 𝐽) |
5 | istopon 21520 | . 2 ⊢ (𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆)) ↔ (𝐽 ∈ Top ∧ (𝑅 Cn 𝑆) = ∪ 𝐽)) | |
6 | 3, 4, 5 | sylanbrc 585 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → 𝐽 ∈ (TopOn‘(𝑅 Cn 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cuni 4838 ‘cfv 6355 (class class class)co 7156 Topctop 21501 TopOnctopon 21518 Cn ccn 21832 ↑ko cxko 22169 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-fin 8513 df-fi 8875 df-rest 16696 df-topgen 16717 df-top 21502 df-topon 21519 df-bases 21554 df-cmp 21995 df-xko 22171 |
This theorem is referenced by: xkoccn 22227 xkopjcn 22264 xkoco1cn 22265 xkoco2cn 22266 xkococn 22268 cnmptkp 22288 cnmptk1 22289 cnmpt1k 22290 cnmptkk 22291 xkofvcn 22292 cnmptk1p 22293 cnmptk2 22294 xkoinjcn 22295 xkocnv 22422 xkohmeo 22423 efmndtmd 22709 symgtgp 22714 |
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