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Theorem ordtrestixx 23137

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 18496	. . . 4 ⊢ ℝ^* = dom ≤
2		letsr 18499	. . . . 5 ⊢ ≤ ∈ TosetRel
3	2	a1i 11	. . . 4 ⊢ (⊤ → ≤ ∈ TosetRel )
4		ordtrestixx.1	. . . . 5 ⊢ 𝐴 ⊆ ℝ^*
5	4	a1i 11	. . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ^*)
6	4	sseli 3925	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ^*)
7	4	sseli 3925	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*)
8		iccval 13284	. . . . . . 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 3965	. . . . 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 23119	. . 3 ⊢ (⊤ → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾_t 𝐴))
14	13	eqcomd 2737	. 2 ⊢ (⊤ → ((ordTop‘ ≤ ) ↾_t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴))))
15	14	mptru 1548	1 ⊢ ((ordTop‘ ≤ ) ↾_t 𝐴) = (ordTop‘( ≤ ∩ (𝐴 × 𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 {crab 3395 ∩ cin 3896 ⊆ wss 3897 class class class wbr 5089 × cxp 5612 ‘cfv 6481 (class class class)co 7346 ℝ^*cxr 11145 ≤ cle 11147 [,]cicc 13248 ↾_t crest 17324 ordTopcordt 17403 TosetRel ctsr 18471

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-1o 8385 df-2o 8386 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fi 9295 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-icc 13252 df-rest 17326 df-topgen 17347 df-ordt 17405 df-ps 18472 df-tsr 18473 df-top 22809 df-topon 22826 df-bases 22861

This theorem is referenced by: ordtresticc 23138 icopnfhmeo 24868

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