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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 13461 | . 2 ⊢ ℝ = ∪ ran (,) | |
2 | retopbas 24721 | . . 3 ⊢ ran (,) ∈ TopBases | |
3 | unitg 22914 | . . 3 ⊢ (ran (,) ∈ TopBases → ∪ (topGen‘ran (,)) = ∪ ran (,)) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ ∪ (topGen‘ran (,)) = ∪ ran (,) |
5 | 1, 4 | eqtr4i 2756 | 1 ⊢ ℝ = ∪ (topGen‘ran (,)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∪ cuni 4909 ran crn 5679 ‘cfv 6549 ℝcr 11139 (,)cioo 13359 topGenctg 17422 TopBasesctb 22892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-ioo 13363 df-topgen 17428 df-bases 22893 |
This theorem is referenced by: retopon 24724 retps 24725 icccld 24727 icopnfcld 24728 iocmnfcld 24729 qdensere 24730 zcld 24773 iccntr 24781 icccmp 24785 retopconn 24789 opnreen 24791 rectbntr0 24792 cnmpopc 24893 evth 24929 evth2 24930 evthicc 25432 ovolicc2 25495 opnmbllem 25574 lhop 25993 dvcnvrelem2 25995 dvcnvre 25996 ftc1 26021 taylthlem2 26354 taylthlem2OLD 26355 ipasslem8 30719 circtopn 33569 tpr2rico 33644 rrhf 33730 rrhqima 33746 rrhre 33753 brsigarn 33934 unibrsiga 33936 sxbrsigalem3 34023 dya2iocucvr 34035 sxbrsigalem1 34036 orrvcval4 34215 orrvcoel 34216 orrvccel 34217 retopsconn 34990 cvmliftlem10 35035 ivthALT 35950 ptrecube 37224 poimirlem29 37253 poimirlem30 37254 poimirlem31 37255 opnmbllem0 37260 mblfinlem1 37261 mblfinlem2 37262 mblfinlem3 37263 mblfinlem4 37264 ismblfin 37265 ftc1cnnc 37296 refsum2cnlem1 44541 sncldre 44548 reopn 44809 ioontr 45034 limciccioolb 45147 limcicciooub 45163 lptre2pt 45166 limclner 45177 limclr 45181 cncfiooicclem1 45419 fperdvper 45445 itgsubsticclem 45501 stoweidlem62 45588 dirkercncflem2 45630 dirkercncflem3 45631 dirkercncflem4 45632 fourierdlem42 45675 fourierdlem58 45690 fourierdlem73 45705 fouriercnp 45752 fouriercn 45758 cnfsmf 46266 incsmf 46268 decsmf 46293 smfpimbor1lem2 46325 |
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