hoissre - Mathbox for Glauco Siliprandi

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Theorem hoissre 46465

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 45100	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))))
5	1	ffvelcdmda 7118	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 8062	. . . 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 8063	. . . . 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 11340	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ^*)
11		icossre 13488	. . 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 583	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))) ⊆ ℝ)
13	4, 12	eqsstrd 4047	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3976 × cxp 5698 ∘ ccom 5704 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 1^st c1st 8028 2^nd c2nd 8029 ℝcr 11183 ℝ^*cxr 11323 [,)cico 13409

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-pre-lttri 11258 ax-pre-lttrn 11259

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-ico 13413

This theorem is referenced by: hoissrrn 46470

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