hoissre - Mathbox for Glauco Siliprandi

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Theorem hoissre 46972

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 45623	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))))
5	1	ffvelcdmda 7036	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ))
6		xp1st 7974	. . . 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 7975	. . . . 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 11195	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2^nd ‘(𝐼‘𝑘)) ∈ ℝ^*)
11		icossre 13381	. . 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 585	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1^st ‘(𝐼‘𝑘))[,)(2^nd ‘(𝐼‘𝑘))) ⊆ ℝ)
13	4, 12	eqsstrd 3956	1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2114 ⊆ wss 3889 × cxp 5629 ∘ ccom 5635 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 1^st c1st 7940 2^nd c2nd 7941 ℝcr 11037 ℝ^*cxr 11178 [,)cico 13300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-1st 7942 df-2nd 7943 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-ico 13304

This theorem is referenced by: hoissrrn 46977

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