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Theorem ordtrestixx 23165

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 18605	. . . 4 ⊢ ℝ^* = dom ≤
2		letsr 18608	. . . . 5 ⊢ ≤ ∈ TosetRel
3	2	a1i 11	. . . 4 ⊢ (⊤ → ≤ ∈ TosetRel )
4		ordtrestixx.1	. . . . 5 ⊢ 𝐴 ⊆ ℝ^*
5	4	a1i 11	. . . 4 ⊢ (⊤ → 𝐴 ⊆ ℝ^*)
6	4	sseli 3959	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ ℝ^*)
7	4	sseli 3959	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ^*)
8		iccval 13406	. . . . . . 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 3999	. . . . 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 23147	. . 3 ⊢ (⊤ → (ordTop‘( ≤ ∩ (𝐴 × 𝐴))) = ((ordTop‘ ≤ ) ↾_t 𝐴))
14	13	eqcomd 2742	. 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 2109 {crab 3420 ∩ cin 3930 ⊆ wss 3931 class class class wbr 5124 × cxp 5657 ‘cfv 6536 (class class class)co 7410 ℝ^*cxr 11273 ≤ cle 11275 [,]cicc 13370 ↾_t crest 17439 ordTopcordt 17518 TosetRel ctsr 18580

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-rep 5254 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-pre-lttri 11208 ax-pre-lttrn 11209

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-int 4928 df-iun 4974 df-iin 4975 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-1o 8485 df-2o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fi 9428 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-icc 13374 df-rest 17441 df-topgen 17462 df-ordt 17520 df-ps 18581 df-tsr 18582 df-top 22837 df-topon 22854 df-bases 22889

This theorem is referenced by: ordtresticc 23166 icopnfhmeo 24897

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