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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 5127 | . . . . . 6 ⊢ ([,) “ (ℝ × ℝ)) = ran ([,) ↾ (ℝ × ℝ)) | |
3 | 1, 2 | eqtri 2644 | . . . . 5 ⊢ 𝐼 = ran ([,) ↾ (ℝ × ℝ)) |
4 | 3 | eleq2i 2693 | . . . 4 ⊢ (𝑋 ∈ 𝐼 ↔ 𝑋 ∈ ran ([,) ↾ (ℝ × ℝ))) |
5 | icoreresf 33200 | . . . . 5 ⊢ ([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ | |
6 | ffn 6045 | . . . . 5 ⊢ (([,) ↾ (ℝ × ℝ)):(ℝ × ℝ)⟶𝒫 ℝ → ([,) ↾ (ℝ × ℝ)) Fn (ℝ × ℝ)) | |
7 | ovelrn 6810 | . . . . 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 264 | . . 3 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏)) |
10 | ovres 6800 | . . . . 5 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎([,) ↾ (ℝ × ℝ))𝑏) = (𝑎[,)𝑏)) | |
11 | 10 | eqeq2d 2632 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏) ↔ 𝑋 = (𝑎[,)𝑏))) |
12 | 11 | 2rexbiia 3055 | . . 3 ⊢ (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎([,) ↾ (ℝ × ℝ))𝑏) ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎[,)𝑏)) |
13 | 9, 12 | bitri 264 | . 2 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎[,)𝑏)) |
14 | icoreval 33201 | . . . 4 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑎[,)𝑏) = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) | |
15 | 14 | eqeq2d 2632 | . . 3 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ∈ ℝ) → (𝑋 = (𝑎[,)𝑏) ↔ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)})) |
16 | 15 | 2rexbiia 3055 | . 2 ⊢ (∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = (𝑎[,)𝑏) ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) |
17 | 13, 16 | bitri 264 | 1 ⊢ (𝑋 ∈ 𝐼 ↔ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ 𝑋 = {𝑧 ∈ ℝ ∣ (𝑎 ≤ 𝑧 ∧ 𝑧 < 𝑏)}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 𝒫 cpw 4158 class class class wbr 4653 × cxp 5112 ran crn 5115 ↾ cres 5116 “ cima 5117 Fn wfn 5883 ⟶wf 5884 (class class class)co 6650 ℝcr 9935 < clt 10074 ≤ cle 10075 [,)cico 12177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ico 12181 |
This theorem is referenced by: isbasisrelowllem1 33203 isbasisrelowllem2 33204 icoreclin 33205 |
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