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Mirrors > Home > MPE Home > Th. List > xkotop | Structured version Visualization version GIF version |
Description: The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
xkotop | ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ ∪ 𝑅 = ∪ 𝑅 | |
2 | eqid 2610 | . . 3 ⊢ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} = {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp} | |
3 | eqid 2610 | . . 3 ⊢ (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | |
4 | 1, 2, 3 | xkoval 21200 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))) |
5 | fibas 20592 | . . 3 ⊢ (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ TopBases | |
6 | tgcl 20584 | . . 3 ⊢ ((fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) ∈ TopBases → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) ∈ Top) | |
7 | 5, 6 | ax-mp 5 | . 2 ⊢ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑅 ∣ (𝑅 ↾t 𝑥) ∈ Comp}, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) ∈ Top |
8 | 4, 7 | syl6eqel 2696 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 {crab 2900 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 ran crn 5039 “ cima 5041 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ficfi 8199 ↾t crest 15904 topGenctg 15921 Topctop 20517 TopBasesctb 20520 Cn ccn 20838 Compccmp 20999 ^ko cxko 21174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 df-er 7629 df-en 7842 df-fin 7845 df-fi 8200 df-topgen 15927 df-top 20521 df-bases 20522 df-xko 21176 |
This theorem is referenced by: xkotopon 21213 xkohaus 21266 xkoptsub 21267 xkococnlem 21272 xkococn 21273 xkohmeo 21428 |
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