![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > iccf | Structured version Visualization version GIF version |
Description: The set of closed intervals of extended reals maps to subsets of extended reals. (Contributed by FL, 14-Jun-2007.) (Revised by Mario Carneiro, 3-Nov-2013.) |
Ref | Expression |
---|---|
iccf | ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-icc 13313 | . 2 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
2 | 1 | ixxf 13316 | 1 ⊢ [,]:(ℝ* × ℝ*)⟶𝒫 ℝ* |
Colors of variables: wff setvar class |
Syntax hints: 𝒫 cpw 4596 × cxp 5667 ⟶wf 6528 ℝ*cxr 11229 ≤ cle 11231 [,]cicc 13309 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-fv 6540 df-oprab 7397 df-mpo 7398 df-1st 7957 df-2nd 7958 df-xr 11234 df-icc 13313 |
This theorem is referenced by: lecldbas 22652 ovolficc 24914 ovolficcss 24915 uniiccdif 25024 uniiccvol 25026 dyadmbllem 25045 dyadmbl 25046 opnmbllem 25047 opnmbllem0 36328 mblfinlem1 36329 mblfinlem2 36330 |
Copyright terms: Public domain | W3C validator |