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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoissre | Structured version Visualization version GIF version |
Description: The projection of a half-open interval onto a single dimension is a subset of ℝ. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
hoissre.1 | ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) |
Ref | Expression |
---|---|
hoissre | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoissre.1 | . . . 4 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
2 | 1 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐼:𝑋⟶(ℝ × ℝ)) |
3 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) | |
4 | 2, 3 | fvovco 43487 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) = ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘)))) |
5 | 1 | ffvelcdmda 7040 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐼‘𝑘) ∈ (ℝ × ℝ)) |
6 | xp1st 7958 | . . . 4 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (1st ‘(𝐼‘𝑘)) ∈ ℝ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (1st ‘(𝐼‘𝑘)) ∈ ℝ) |
8 | xp2nd 7959 | . . . . 5 ⊢ ((𝐼‘𝑘) ∈ (ℝ × ℝ) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ) | |
9 | 5, 8 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ) |
10 | 9 | rexrd 11212 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (2nd ‘(𝐼‘𝑘)) ∈ ℝ*) |
11 | icossre 13352 | . . 3 ⊢ (((1st ‘(𝐼‘𝑘)) ∈ ℝ ∧ (2nd ‘(𝐼‘𝑘)) ∈ ℝ*) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) ⊆ ℝ) | |
12 | 7, 10, 11 | syl2anc 585 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((1st ‘(𝐼‘𝑘))[,)(2nd ‘(𝐼‘𝑘))) ⊆ ℝ) |
13 | 4, 12 | eqsstrd 3987 | 1 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3915 × cxp 5636 ∘ ccom 5642 ⟶wf 6497 ‘cfv 6501 (class class class)co 7362 1st c1st 7924 2nd c2nd 7925 ℝcr 11057 ℝ*cxr 11195 [,)cico 13273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-pre-lttri 11132 ax-pre-lttrn 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-po 5550 df-so 5551 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-1st 7926 df-2nd 7927 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-ico 13277 |
This theorem is referenced by: hoissrrn 44864 |
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