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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 6919 | . . . . 5 ⊢ (([,) ∘ 𝐼)‘𝑘) ∈ V | |
| 2 | 1 | rgenw 3065 | . . . 4 ⊢ ∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ∈ V |
| 3 | ixpssmapg 8968 | . . . 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 11246 | . . . 4 ⊢ ℝ ∈ V | |
| 7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → ℝ ∈ V) |
| 8 | hoissrrn.1 | . . . . . 6 ⊢ (𝜑 → 𝐼:𝑋⟶(ℝ × ℝ)) | |
| 9 | 8 | hoissre 46559 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
| 10 | 9 | ralrimiva 3146 | . . . 4 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
| 11 | iunss 5045 | . . . 4 ⊢ (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) | |
| 12 | 10, 11 | sylibr 234 | . . 3 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) |
| 13 | mapss 8929 | . . 3 ⊢ ((ℝ ∈ V ∧ ∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ ℝ) → (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)) | |
| 14 | 7, 12, 13 | syl2anc 584 | . 2 ⊢ (𝜑 → (∪ 𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ↑m 𝑋) ⊆ (ℝ ↑m 𝑋)) |
| 15 | 5, 14 | sstrd 3994 | 1 ⊢ (𝜑 → X𝑘 ∈ 𝑋 (([,) ∘ 𝐼)‘𝑘) ⊆ (ℝ ↑m 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ∪ ciun 4991 × cxp 5683 ∘ ccom 5689 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Xcixp 8937 ℝcr 11154 [,)cico 13389 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 |
| This theorem is referenced by: ovnlecvr 46573 |
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