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Mirrors > Home > MPE Home > Th. List > areambl | Structured version Visualization version GIF version |
Description: The fibers of a measurable region are finitely measurable subsets of ℝ. (Contributed by Mario Carneiro, 21-Jun-2015.) |
Ref | Expression |
---|---|
areambl | ⊢ ((𝑆 ∈ dom area ∧ 𝐴 ∈ ℝ) → ((𝑆 “ {𝐴}) ∈ dom vol ∧ (vol‘(𝑆 “ {𝐴})) ∈ ℝ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmarea 26330 | . . . 4 ⊢ (𝑆 ∈ dom area ↔ (𝑆 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿1)) | |
2 | 1 | simp2bi 1147 | . . 3 ⊢ (𝑆 ∈ dom area → ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (◡vol “ ℝ)) |
3 | sneq 4600 | . . . . . 6 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
4 | 3 | imaeq2d 6017 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑆 “ {𝑥}) = (𝑆 “ {𝐴})) |
5 | 4 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑆 “ {𝑥}) ∈ (◡vol “ ℝ) ↔ (𝑆 “ {𝐴}) ∈ (◡vol “ ℝ))) |
6 | 5 | rspccva 3582 | . . 3 ⊢ ((∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (◡vol “ ℝ) ∧ 𝐴 ∈ ℝ) → (𝑆 “ {𝐴}) ∈ (◡vol “ ℝ)) |
7 | 2, 6 | sylan 581 | . 2 ⊢ ((𝑆 ∈ dom area ∧ 𝐴 ∈ ℝ) → (𝑆 “ {𝐴}) ∈ (◡vol “ ℝ)) |
8 | volf 24916 | . . 3 ⊢ vol:dom vol⟶(0[,]+∞) | |
9 | ffn 6672 | . . 3 ⊢ (vol:dom vol⟶(0[,]+∞) → vol Fn dom vol) | |
10 | elpreima 7012 | . . 3 ⊢ (vol Fn dom vol → ((𝑆 “ {𝐴}) ∈ (◡vol “ ℝ) ↔ ((𝑆 “ {𝐴}) ∈ dom vol ∧ (vol‘(𝑆 “ {𝐴})) ∈ ℝ))) | |
11 | 8, 9, 10 | mp2b 10 | . 2 ⊢ ((𝑆 “ {𝐴}) ∈ (◡vol “ ℝ) ↔ ((𝑆 “ {𝐴}) ∈ dom vol ∧ (vol‘(𝑆 “ {𝐴})) ∈ ℝ)) |
12 | 7, 11 | sylib 217 | 1 ⊢ ((𝑆 ∈ dom area ∧ 𝐴 ∈ ℝ) → ((𝑆 “ {𝐴}) ∈ dom vol ∧ (vol‘(𝑆 “ {𝐴})) ∈ ℝ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ⊆ wss 3914 {csn 4590 ↦ cmpt 5192 × cxp 5635 ◡ccnv 5636 dom cdm 5637 “ cima 5640 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 0cc0 11059 +∞cpnf 11194 [,]cicc 13276 volcvol 24850 𝐿1cibl 25004 areacarea 26328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-ico 13279 df-icc 13280 df-fz 13434 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-sum 15580 df-ovol 24851 df-vol 24852 df-itg 25010 df-area 26329 |
This theorem is referenced by: areaf 26334 |
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