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| Mirrors > Home > MPE Home > Th. List > ordtrestixx | Structured version Visualization version GIF version | ||
| Description: The restriction of the less than order to an interval gives the same topology as the subspace topology. (Contributed by Mario Carneiro, 9-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| ordtrestixx.1 | ⊢ 𝐴 ⊆ ℝ* | 
| ordtrestixx.2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) | 
| Ref | Expression | 
|---|---|
| ordtrestixx | ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ledm 18635 | . . . 4 ⊢ ℝ* = dom ≤ | |
| 2 | letsr 18638 | . . . . 5 ⊢ ≤ ∈ TosetRel | |
| 3 | 2 | a1i 11 | . . . 4 ⊢ (⊤ → ≤ ∈ TosetRel ) | 
| 4 | ordtrestixx.1 | . . . . 5 ⊢ 𝐴 ⊆ ℝ* | |
| 5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ*) | 
| 6 | 4 | sseli 3979 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ*) | 
| 7 | 4 | sseli 3979 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ*) | 
| 8 | iccval 13426 | . . . . . . 7 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
| 9 | 6, 7, 8 | syl2an 596 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | 
| 10 | ordtrestixx.2 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴) | |
| 11 | 9, 10 | eqsstrrd 4019 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) | 
| 12 | 11 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴) | 
| 13 | 1, 3, 5, 12 | ordtrest2 23212 | . . 3 ⊢ (⊤ → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾t 𝐴)) | 
| 14 | 13 | eqcomd 2743 | . 2 ⊢ (⊤ → ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))) | 
| 15 | 14 | mptru 1547 | 1 ⊢ ((ordTop‘ ≤ ) ↾t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2108 {crab 3436 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ℝ*cxr 11294 ≤ cle 11296 [,]cicc 13390 ↾t crest 17465 ordTopcordt 17544 TosetRel ctsr 18610 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-icc 13394 df-rest 17467 df-topgen 17488 df-ordt 17546 df-ps 18611 df-tsr 18612 df-top 22900 df-topon 22917 df-bases 22953 | 
| This theorem is referenced by: ordtresticc 23231 icopnfhmeo 24974 | 
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