hoissre - Mathbox for Glauco Siliprandi

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Theorem hoissre 46559

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 45198	. 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 8046	. . . 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 8047	. . . . 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 11311	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ^*)
11		icossre 13468	. . 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 4018	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3951 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 ℝcr 11154 ℝ^*cxr 11294 [,)cico 13389

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393

This theorem is referenced by: hoissrrn 46564

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