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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 6823 | . . . . 5 ⊢ (𝑋 ↾t ℝ) = ((ordTop‘ ≤ ) ↾t ℝ) |
4 | 3 | xrtgioo 22810 | . . . 4 ⊢ (topGen‘ran (,)) = (𝑋 ↾t ℝ) |
5 | 1, 4 | eqtri 2782 | . . 3 ⊢ 𝑅 = (𝑋 ↾t ℝ) |
6 | 5 | oveq1i 6823 | . 2 ⊢ (𝑅 ↾t 𝐴) = ((𝑋 ↾t ℝ) ↾t 𝐴) |
7 | fvex 6362 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
8 | 2, 7 | eqeltri 2835 | . . 3 ⊢ 𝑋 ∈ V |
9 | reex 10219 | . . 3 ⊢ ℝ ∈ V | |
10 | restabs 21171 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴 ⊆ ℝ ∧ ℝ ∈ V) → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) | |
11 | 8, 9, 10 | mp3an13 1564 | . 2 ⊢ (𝐴 ⊆ ℝ → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) |
12 | 6, 11 | syl5req 2807 | 1 ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 Vcvv 3340 ⊆ wss 3715 ran crn 5267 ‘cfv 6049 (class class class)co 6813 ℝcr 10127 ≤ cle 10267 (,)cioo 12368 ↾t crest 16283 topGenctg 16300 ordTopcordt 16361 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fi 8482 df-sup 8513 df-inf 8514 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-n0 11485 df-z 11570 df-uz 11880 df-q 11982 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-rest 16285 df-topgen 16306 df-ordt 16363 df-ps 17401 df-tsr 17402 df-top 20901 df-topon 20918 df-bases 20952 |
This theorem is referenced by: xrrest2 22812 dfii5 22889 |
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