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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 | β’ ((π β Top β§ π β Top) β (π βko π ) β Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 β’ βͺ π = βͺ π | |
2 | eqid 2736 | . . 3 β’ {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp} = {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp} | |
3 | eqid 2736 | . . 3 β’ (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£}) = (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£}) | |
4 | 1, 2, 3 | xkoval 22844 | . 2 β’ ((π β Top β§ π β Top) β (π βko π ) = (topGenβ(fiβran (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£})))) |
5 | fibas 22233 | . . 3 β’ (fiβran (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£})) β TopBases | |
6 | tgcl 22225 | . . 3 β’ ((fiβran (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£})) β TopBases β (topGenβ(fiβran (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£}))) β Top) | |
7 | 5, 6 | ax-mp 5 | . 2 β’ (topGenβ(fiβran (π β {π₯ β π« βͺ π β£ (π βΎt π₯) β Comp}, π£ β π β¦ {π β (π Cn π) β£ (π β π) β π£}))) β Top |
8 | 4, 7 | eqeltrdi 2845 | 1 β’ ((π β Top β§ π β Top) β (π βko π ) β Top) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 β wcel 2105 {crab 3403 β wss 3898 π« cpw 4547 βͺ cuni 4852 ran crn 5621 β cima 5623 βcfv 6479 (class class class)co 7337 β cmpo 7339 ficfi 9267 βΎt crest 17228 topGenctg 17245 Topctop 22148 TopBasesctb 22201 Cn ccn 22481 Compccmp 22643 βko cxko 22818 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-int 4895 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-en 8805 df-fin 8808 df-fi 9268 df-topgen 17251 df-top 22149 df-bases 22202 df-xko 22820 |
This theorem is referenced by: xkotopon 22857 xkohaus 22910 xkoptsub 22911 xkococnlem 22916 xkococn 22917 xkohmeo 23072 |
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