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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > icof | Structured version Visualization version GIF version |
Description: The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
icof | ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2736 | . . . 4 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} = {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
2 | ssrab2 4090 | . . . . 5 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ* | |
3 | xrex 13027 | . . . . . . 7 ⊢ ℝ* ∈ V | |
4 | 3 | rabex 5345 | . . . . . 6 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ V |
5 | 4 | elpw 4609 | . . . . 5 ⊢ ({𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ⊆ ℝ*) |
6 | 2, 5 | mpbir 231 | . . . 4 ⊢ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* |
7 | 1, 6 | eqeltrrdi 2848 | . . 3 ⊢ ((𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ*) → {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ*) |
8 | 7 | rgen2 3197 | . 2 ⊢ ∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* |
9 | df-ico 13390 | . . 3 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
10 | 9 | fmpo 8092 | . 2 ⊢ (∀𝑥 ∈ ℝ* ∀𝑦 ∈ ℝ* {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)} ∈ 𝒫 ℝ* ↔ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*) |
11 | 8, 10 | mpbi 230 | 1 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 𝒫 cpw 4605 class class class wbr 5148 × cxp 5687 ⟶wf 6559 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 [,)cico 13386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-xr 11297 df-ico 13390 |
This theorem is referenced by: fvvolicof 45947 volicoff 45951 voliooicof 45952 ovolval5lem2 46609 ovolval5lem3 46610 ovnovollem1 46612 ovnovollem2 46613 |
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