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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 24782 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
| 2 | uniretop 24783 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
| 3 | 2 | toptopon 22923 | . 2 ⊢ ((topGen‘ran (,)) ∈ Top ↔ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) |
| 4 | 1, 3 | mpbi 230 | 1 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ran crn 5686 ‘cfv 6561 ℝcr 11154 (,)cioo 13387 topGenctg 17482 Topctop 22899 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-top 22900 df-topon 22917 df-bases 22953 |
| This theorem is referenced by: xrtgioo 24828 reconnlem1 24848 reconn 24850 cnmpopc 24955 cnrehmeo 24984 cnrehmeoOLD 24985 bndth 24990 evth2 24992 htpycc 25012 pcocn 25050 pcohtpylem 25052 pcopt 25055 pcopt2 25056 pcoass 25057 pcorevlem 25059 circcn 33837 tpr2tp 33903 sxbrsiga 34292 cvmliftlem8 35297 knoppcnlem10 36503 knoppcnlem11 36504 poimir 37660 broucube 37661 cnambfre 37675 reheibor 37846 rfcnpre1 45024 fcnre 45030 refsumcn 45035 refsum2cnlem1 45042 climreeq 45628 islptre 45634 icccncfext 45902 stoweidlem47 46062 dirkercncflem4 46121 dirkercncf 46122 fourierdlem62 46183 |
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