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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hoissrrn | Structured version Visualization version GIF version |
Description: A half-open interval is a subset of R^n . (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
hoissrrn.1 | ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) |
Ref | Expression |
---|---|
hoissrrn | ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑𝑚 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6510 | . . . . 5 ⊢ (([,) ∘ 𝐼)‘𝑘) ∈ V | |
2 | 1 | rgenw 3095 | . . . 4 ⊢ ∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ∈ V |
3 | ixpssmapg 8288 | . . . 4 ⊢ (∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ∈ V → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑𝑚 𝑋)) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑𝑚 𝑋) |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑𝑚 𝑋)) |
6 | reex 10425 | . . . 4 ⊢ ℝ ∈ V | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℝ ∈ V) |
8 | hoissrrn.1 | . . . . . 6 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
9 | 8 | hoissre 42287 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
10 | 9 | ralrimiva 3127 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
11 | iunss 4832 | . . . 4 ⊢ (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) | |
12 | 10, 11 | sylibr 226 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
13 | mapss 8250 | . . 3 ⊢ ((ℝ ∈ V ∧ ∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) → (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)) | |
14 | 7, 12, 13 | syl2anc 576 | . 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑𝑚 𝑋) ⊆ (ℝ ↑𝑚 𝑋)) |
15 | 5, 14 | sstrd 3863 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑𝑚 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2051 ∀wral 3083 Vcvv 3410 ⊆ wss 3824 ∪ ciun 4789 × cxp 5402 ∘ ccom 5408 ⟶wf 6182 ‘cfv 6186 (class class class)co 6975 ↑𝑚 cmap 8205 Xcixp 8258 ℝcr 10333 [,)cico 12555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-rep 5046 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-pre-lttri 10408 ax-pre-lttrn 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-ov 6978 df-oprab 6979 df-mpo 6980 df-1st 7500 df-2nd 7501 df-er 8088 df-map 8207 df-ixp 8259 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-ico 12559 |
This theorem is referenced by: ovnlecvr 42301 |
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