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Theorem ordtrestixx 22655

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 18525	. . . 4 ⊢ ℝ^* = dom ≤
2		letsr 18528	. . . . 5 ⊢ ≤ ∈ TosetRel
3	2	a1i 11	. . . 4 ⊢ (⊤ → ≤ ∈ TosetRel )
4		ordtrestixx.1	. . . . 5 ⊢ 𝐴 ⊆ ℝ^*
5	4	a1i 11	. . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ^*)
6	4	sseli 3974	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ^*)
7	4	sseli 3974	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*)
8		iccval 13345	. . . . . . 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 4017	. . . . 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 22637	. . 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 396 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 {crab 3431 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5141 × cxp 5667 ‘cfv 6532 (class class class)co 7393 ℝ^*cxr 11229 ≤ cle 11231 [,]cicc 13309 ↾_t crest 17348 ordTopcordt 17427 TosetRel ctsr 18500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-pre-lttri 11166 ax-pre-lttrn 11167

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-1st 7957 df-2nd 7958 df-1o 8448 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fi 9388 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-icc 13313 df-rest 17350 df-topgen 17371 df-ordt 17429 df-ps 18501 df-tsr 18502 df-top 22325 df-topon 22342 df-bases 22378

This theorem is referenced by: ordtresticc 22656 icopnfhmeo 24388

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