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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 22942 | . 2 ⊢ (topGen‘ran (,)) ∈ Top | |
2 | uniretop 22943 | . . 3 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
3 | 2 | toptopon 21099 | . 2 ⊢ ((topGen‘ran (,)) ∈ Top ↔ (topGen‘ran (,)) ∈ (TopOn‘ℝ)) |
4 | 1, 3 | mpbi 222 | 1 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2164 ran crn 5347 ‘cfv 6127 ℝcr 10258 (,)cioo 12470 topGenctg 16458 Topctop 21075 TopOnctopon 21092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-pre-lttri 10333 ax-pre-lttrn 10334 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-ioo 12474 df-topgen 16464 df-top 21076 df-topon 21093 df-bases 21128 |
This theorem is referenced by: xrtgioo 22986 reconnlem1 23006 reconn 23008 cnmpt2pc 23104 cnrehmeo 23129 bndth 23134 evth2 23136 htpycc 23156 pcocn 23193 pcohtpylem 23195 pcopt 23198 pcopt2 23199 pcoass 23200 pcorevlem 23202 circcn 30446 tpr2tp 30491 sxbrsiga 30893 cvmliftlem8 31816 knoppcnlem10 33020 knoppcnlem11 33021 poimir 33981 broucube 33982 cnambfre 33996 reheibor 34175 rfcnpre1 39991 fcnre 39997 refsumcn 40002 refsum2cnlem1 40009 climreeq 40634 islptre 40640 icccncfext 40889 stoweidlem47 41052 dirkercncflem4 41111 dirkercncf 41112 fourierdlem62 41173 |
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