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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 6559 | . . . . 5 ⊢ (𝑋 ↾t ℝ) = ((ordTop‘ ≤ ) ↾t ℝ) |
4 | 3 | xrtgioo 22417 | . . . 4 ⊢ (topGen‘ran (,)) = (𝑋 ↾t ℝ) |
5 | 1, 4 | eqtri 2632 | . . 3 ⊢ 𝑅 = (𝑋 ↾t ℝ) |
6 | 5 | oveq1i 6559 | . 2 ⊢ (𝑅 ↾t 𝐴) = ((𝑋 ↾t ℝ) ↾t 𝐴) |
7 | fvex 6113 | . . . 4 ⊢ (ordTop‘ ≤ ) ∈ V | |
8 | 2, 7 | eqeltri 2684 | . . 3 ⊢ 𝑋 ∈ V |
9 | reex 9906 | . . 3 ⊢ ℝ ∈ V | |
10 | restabs 20779 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴 ⊆ ℝ ∧ ℝ ∈ V) → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) | |
11 | 8, 9, 10 | mp3an13 1407 | . 2 ⊢ (𝐴 ⊆ ℝ → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) |
12 | 6, 11 | syl5req 2657 | 1 ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 ≤ cle 9954 (,)cioo 12046 ↾t crest 15904 topGenctg 15921 ordTopcordt 15982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-rest 15906 df-topgen 15927 df-ordt 15984 df-ps 17023 df-tsr 17024 df-top 20521 df-bases 20522 df-topon 20523 |
This theorem is referenced by: xrrest2 22419 dfii5 22496 |
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