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Mirrors > Home > MPE Home > Th. List > Mathboxes > icoreelrnab | Structured version Visualization version GIF version |
Description: Elementhood in the set of closed-below, open-above intervals of reals. (Contributed by ML, 27-Jul-2020.) |
Ref | Expression |
---|---|
icoreelrnab.1 | ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) |
Ref | Expression |
---|---|
icoreelrnab | ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | icoreelrnab.1 | . . . . . 6 ⊢ 𝐼 = ([,) “ (ℝ × ℝ)) | |
2 | df-ima 5532 | . . . . . 6 ⊢ ([,) “ (ℝ × ℝ)) = ran ([,) ↾ (ℝ × ℝ)) | |
3 | 1, 2 | eqtri 2821 | . . . . 5 ⊢ 𝐼 = ran ([,) ↾ (ℝ × ℝ)) |
4 | 3 | eleq2i 2881 | . . . 4 ⊢ (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ ran ([,) ↾ (ℝ × ℝ))) |
5 | icoreresf 34769 | . . . . 5 ⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ | |
6 | ffn 6487 | . . . . 5 ⊢ (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ → ([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
7 | ovelrn 7304 | . . . . 5 ⊢ (([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ) → (𝑋 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏))) | |
8 | 5, 6, 7 | mp2b 10 | . . . 4 ⊢ (𝑋 ∈ ran ([,) ↾ (ℝ × ℝ)) ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏)) |
9 | 4, 8 | bitri 278 | . . 3 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏)) |
10 | ovres 7294 | . . . . 5 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎([,) ↾ (ℝ × ℝ))𝑏) = (𝑎[,)𝑏)) | |
11 | 10 | eqeq2d 2809 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏) ↔ 𝑋 = (𝑎[,)𝑏))) |
12 | 11 | 2rexbiia 3257 | . . 3 ⊢ (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏) ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎[,)𝑏)) |
13 | 9, 12 | bitri 278 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎[,)𝑏)) |
14 | icoreval 34770 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎[,)𝑏) = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | |
15 | 14 | eqeq2d 2809 | . . 3 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑋 = (𝑎[,)𝑏) ↔ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) |
16 | 15 | 2rexbiia 3257 | . 2 ⊢ (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎[,)𝑏) ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) |
17 | 13, 16 | bitri 278 | 1 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 𝒫 cpw 4497 class class class wbr 5030 × cxp 5517 ran crn 5520 ↾ cres 5521 “ cima 5522 Fn wfn 6319 ⟶wf 6320 (class class class)co 7135 ℝcr 10525 < clt 10664 ≤ cle 10665 [,)cico 12728 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ico 12732 |
This theorem is referenced by: isbasisrelowllem1 34772 isbasisrelowllem2 34773 icoreclin 34774 |
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