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| Mirrors > Home > MPE Home > Th. List > uniretop | Structured version Visualization version GIF version | ||
| Description: The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.) |
| Ref | Expression |
|---|---|
| uniretop | ⊢ ℝ = ∪ (topGen‘ran (,)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unirnioo 13489 | . 2 ⊢ ℝ = ∪ ran (,) | |
| 2 | retopbas 24781 | . . 3 ⊢ ran (,) ∈ TopBases | |
| 3 | unitg 22974 | . . 3 ⊢ (ran (,) ∈ TopBases → ∪ (topGen‘ran (,)) = ∪ ran (,)) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ ∪ (topGen‘ran (,)) = ∪ ran (,) |
| 5 | 1, 4 | eqtr4i 2768 | 1 ⊢ ℝ = ∪ (topGen‘ran (,)) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 ℝcr 11154 (,)cioo 13387 topGenctg 17482 TopBasesctb 22952 |
| 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-bases 22953 |
| This theorem is referenced by: retopon 24784 retps 24785 icccld 24787 icopnfcld 24788 iocmnfcld 24789 qdensere 24790 zcld 24835 iccntr 24843 icccmp 24847 retopconn 24851 opnreen 24853 rectbntr0 24854 cnmpopc 24955 evth 24991 evth2 24992 evthicc 25494 ovolicc2 25557 opnmbllem 25636 lhop 26055 dvcnvrelem2 26057 dvcnvre 26058 ftc1 26083 taylthlem2 26416 taylthlem2OLD 26417 ipasslem8 30856 circtopn 33836 tpr2rico 33911 rrhf 33999 rrhqima 34015 rrhre 34022 brsigarn 34185 unibrsiga 34187 sxbrsigalem3 34274 dya2iocucvr 34286 sxbrsigalem1 34287 orrvcval4 34467 orrvcoel 34468 orrvccel 34469 retopsconn 35254 cvmliftlem10 35299 ivthALT 36336 ptrecube 37627 poimirlem29 37656 poimirlem30 37657 poimirlem31 37658 opnmbllem0 37663 mblfinlem1 37664 mblfinlem2 37665 mblfinlem3 37666 mblfinlem4 37667 ismblfin 37668 ftc1cnnc 37699 readvrec2 42391 refsum2cnlem1 45042 sncldre 45049 reopn 45301 ioontr 45524 limciccioolb 45636 limcicciooub 45652 lptre2pt 45655 limclner 45666 limclr 45670 cncfiooicclem1 45908 fperdvper 45934 itgsubsticclem 45990 stoweidlem62 46077 dirkercncflem2 46119 dirkercncflem3 46120 dirkercncflem4 46121 fourierdlem42 46164 fourierdlem58 46179 fourierdlem73 46194 fouriercnp 46241 fouriercn 46247 cnfsmf 46755 incsmf 46757 decsmf 46782 smfpimbor1lem2 46814 |
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