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Mirrors > Home > MPE Home > Th. List > xrrest | Structured version Visualization version GIF version |
Description: The subspace topology induced by a subset of the reals. (Contributed by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
xrrest.1 | ⊢ 𝑋 = (ordTop‘ ≤ ) |
xrrest.2 | ⊢ 𝑅 = (topGen‘ran (,)) |
Ref | Expression |
---|---|
xrrest | ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrrest.2 | . . . 4 ⊢ 𝑅 = (topGen‘ran (,)) | |
2 | xrrest.1 | . . . . . 6 ⊢ 𝑋 = (ordTop‘ ≤ ) | |
3 | 2 | oveq1i 7155 | . . . . 5 ⊢ (𝑋 ↾t ℝ) = ((ordTop‘ ≤ ) ↾t ℝ) |
4 | 3 | xrtgioo 23341 | . . . 4 ⊢ (topGen‘ran (,)) = (𝑋 ↾t ℝ) |
5 | 1, 4 | eqtri 2841 | . . 3 ⊢ 𝑅 = (𝑋 ↾t ℝ) |
6 | 5 | oveq1i 7155 | . 2 ⊢ (𝑅 ↾t 𝐴) = ((𝑋 ↾t ℝ) ↾t 𝐴) |
7 | 2 | fvexi 6677 | . . 3 ⊢ 𝑋 ∈ V |
8 | reex 10616 | . . 3 ⊢ ℝ ∈ V | |
9 | restabs 21701 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴 ⊆ ℝ ∧ ℝ ∈ V) → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) | |
10 | 7, 8, 9 | mp3an13 1443 | . 2 ⊢ (𝐴 ⊆ ℝ → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) |
11 | 6, 10 | syl5req 2866 | 1 ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 ≤ cle 10664 (,)cioo 12726 ↾t crest 16682 topGenctg 16699 ordTopcordt 16760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fi 8863 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-rest 16684 df-topgen 16705 df-ordt 16762 df-ps 17798 df-tsr 17799 df-top 21430 df-topon 21447 df-bases 21482 |
This theorem is referenced by: xrrest2 23343 dfii5 23420 |
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