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Mirrors > Home > MPE Home > Th. List > areass | Structured version Visualization version GIF version |
Description: A measurable region is a subset of ℝ × ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
areass | ⊢ (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmarea 24684 | . 2 ⊢ (𝑆 ∈ dom area ↔ (𝑆 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿1)) | |
2 | 1 | simp1bi 1076 | 1 ⊢ (𝑆 ∈ dom area → 𝑆 ⊆ (ℝ × ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 {csn 4177 ↦ cmpt 4729 × cxp 5112 ◡ccnv 5113 dom cdm 5114 “ cima 5117 ‘cfv 5888 ℝcr 9935 volcvol 23232 𝐿1cibl 23386 areacarea 24682 |
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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-fv 5896 df-sum 14417 df-itg 23392 df-area 24683 |
This theorem is referenced by: (None) |
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