Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7201 | . . . . 5 ⊢ (𝑋 ↾t ℝ) = ((ordTop‘ ≤ ) ↾t ℝ) |
4 | 3 | xrtgioo 23657 | . . . 4 ⊢ (topGen‘ran (,)) = (𝑋 ↾t ℝ) |
5 | 1, 4 | eqtri 2759 | . . 3 ⊢ 𝑅 = (𝑋 ↾t ℝ) |
6 | 5 | oveq1i 7201 | . 2 ⊢ (𝑅 ↾t 𝐴) = ((𝑋 ↾t ℝ) ↾t 𝐴) |
7 | 2 | fvexi 6709 | . . 3 ⊢ 𝑋 ∈ V |
8 | reex 10785 | . . 3 ⊢ ℝ ∈ V | |
9 | restabs 22016 | . . 3 ⊢ ((𝑋 ∈ V ∧ 𝐴 ⊆ ℝ ∧ ℝ ∈ V) → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) | |
10 | 7, 8, 9 | mp3an13 1454 | . 2 ⊢ (𝐴 ⊆ ℝ → ((𝑋 ↾t ℝ) ↾t 𝐴) = (𝑋 ↾t 𝐴)) |
11 | 6, 10 | eqtr2id 2784 | 1 ⊢ (𝐴 ⊆ ℝ → (𝑋 ↾t 𝐴) = (𝑅 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ⊆ wss 3853 ran crn 5537 ‘cfv 6358 (class class class)co 7191 ℝcr 10693 ≤ cle 10833 (,)cioo 12900 ↾t crest 16879 topGenctg 16896 ordTopcordt 16958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fi 9005 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-q 12510 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-rest 16881 df-topgen 16902 df-ordt 16960 df-ps 18026 df-tsr 18027 df-top 21745 df-topon 21762 df-bases 21797 |
This theorem is referenced by: xrrest2 23659 dfii5 23736 |
Copyright terms: Public domain | W3C validator |