hoissre - Mathbox for Glauco Siliprandi

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Theorem hoissre 46987

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 481	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ))
3		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
4	2, 3	fvovco 45640	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))))
5	1	ffvelcdmda 7025	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 7963	. . . 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 7964	. . . . 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 11186	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ^*)
11		icossre 13372	. . 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 590	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))) ⊆ ℝ)
13	4, 12	eqsstrd 3949	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2119 ⊆ wss 3883 × cxp 5616 ∘ ccom 5622 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 ℝcr 11028 ℝ^*cxr 11169 [,)cico 13291

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-pre-lttri 11103 ax-pre-lttrn 11104

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-po 5526 df-so 5527 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-ico 13295

This theorem is referenced by: hoissrrn 46992

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