hoissre - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hoissre		Structured version Visualization version GIF version

Theorem hoissre 46500

Description: The projection of a half-open interval onto a single dimension is a subset of ℝ. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

**Hypothesis**
Ref	Expression
hoissre.1	⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))

**Assertion**
Ref	Expression
hoissre	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)

Distinct variable group: 𝑘,𝑋 Allowed substitution hints: 𝜑(𝑘) 𝐼(𝑘)

**Proof of Theorem hoissre**
Step	Hyp	Ref	Expression
1		hoissre.1	. . . 4 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ))
2	1	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
4	2, 3	fvovco 45136	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))))
5	1	ffvelcdmda 7104	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 8045	. . . 4 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (1^st ‘(𝐼‘𝑘)) ∈ ℝ)
7	5, 6	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1^st ‘(𝐼‘𝑘)) ∈ ℝ)
8		xp2nd 8046	. . . . 5 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ)
9	5, 8	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ)
10	9	rexrd 11309	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ^*)
11		icossre 13465	. . 3 ⊢ (((1^st ‘(𝐼‘𝑘)) ∈ ℝ ∧ (2^nd ‘(𝐼‘𝑘)) ∈ ℝ^*) → ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))) ⊆ ℝ)
12	7, 10, 11	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))) ⊆ ℝ)
13	4, 12	eqsstrd 4034	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ⊆ wss 3963 × cxp 5687 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 ℝcr 11152 ℝ^*cxr 11292 [,)cico 13386

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ico 13390

This theorem is referenced by: hoissrrn 46505

W3C validator