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Theorem ordtrestixx 22373

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.)

**Hypotheses**
Ref	Expression
ordtrestixx.1	⊢ 𝐴 ⊆ ℝ^*
ordtrestixx.2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥[,]𝑦) ⊆ 𝐴)

**Assertion**
Ref	Expression
ordtrestixx	⊢ ((ordTop‘ ≤ ) ↾_t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))

Distinct variable group: 𝑥,𝑦,𝐴

Proof of Theorem ordtrestixx Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ledm 18308	. . . 4 ⊢ ℝ^* = dom ≤
2		letsr 18311	. . . . 5 ⊢ ≤ ∈ TosetRel
3	2	a1i 11	. . . 4 ⊢ (⊤ → ≤ ∈ TosetRel )
4		ordtrestixx.1	. . . . 5 ⊢ 𝐴 ⊆ ℝ^*
5	4	a1i 11	. . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ^*)
6	4	sseli 3917	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ^*)
7	4	sseli 3917	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*)
8		iccval 13118	. . . . . . 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 3960	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)
12	11	adantl 482	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)} ⊆ 𝐴)
13	1, 3, 5, 12	ordtrest2 22355	. . 3 ⊢ (⊤ → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾_t 𝐴))
14	13	eqcomd 2744	. 2 ⊢ (⊤ → ((ordTop‘ ≤ ) ↾_t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
15	14	mptru 1546	1 ⊢ ((ordTop‘ ≤ ) ↾_t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ⊤wtru 1540 ∈ wcel 2106 {crab 3068 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ℝ^*cxr 11008 ≤ cle 11010 [,]cicc 13082 ↾_t crest 17131 ordTopcordt 17210 TosetRel ctsr 18283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fi 9170 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-icc 13086 df-rest 17133 df-topgen 17154 df-ordt 17212 df-ps 18284 df-tsr 18285 df-top 22043 df-topon 22060 df-bases 22096

This theorem is referenced by: ordtresticc 22374 icopnfhmeo 24106

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