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Mirrors > Home > MPE Home > Th. List > retopon | Structured version Visualization version GIF version |
Description: The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
retopon | ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | retop 24769 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | uniretop 24770 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
3 | 2 | toptopon 22910 | . 2 ⊢ ((topGen‘ran (,)) ∈ Top ↔ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) |
4 | 1, 3 | mpbi 229 | 1 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 ran crn 5683 ‘cfv 6554 ℝcr 11157 (,)cioo 13378 topGenctg 17452 Topctop 22886 TopOnctopon 22903 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-ioo 13382 df-topgen 17458 df-top 22887 df-topon 22904 df-bases 22940 |
This theorem is referenced by: xrtgioo 24813 reconnlem1 24833 reconn 24835 cnmpopc 24940 cnrehmeo 24969 cnrehmeoOLD 24970 bndth 24975 evth2 24977 htpycc 24997 pcocn 25035 pcohtpylem 25037 pcopt 25040 pcopt2 25041 pcoass 25042 pcorevlem 25044 circcn 33653 tpr2tp 33719 sxbrsiga 34124 cvmliftlem8 35120 knoppcnlem10 36205 knoppcnlem11 36206 poimir 37354 broucube 37355 cnambfre 37369 reheibor 37540 rfcnpre1 44618 fcnre 44624 refsumcn 44629 refsum2cnlem1 44636 climreeq 45234 islptre 45240 icccncfext 45508 stoweidlem47 45668 dirkercncflem4 45727 dirkercncf 45728 fourierdlem62 45789 |
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