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Mirrors > Home > MPE Home > Th. List > i1fima | Structured version Visualization version GIF version |
Description: Any preimage of a simple function is measurable. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
i1fima | ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1ff 24204 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
2 | ffun 6510 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹) | |
3 | inpreima 6826 | . . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
4 | iunid 4975 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦} = (𝐴 ∩ ran 𝐹) | |
5 | 4 | imaeq2i 5920 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)) |
6 | imaiun 6995 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) | |
7 | 5, 6 | eqtr3i 2843 | . . . 4 ⊢ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) |
8 | cnvimass 5942 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
9 | cnvimarndm 5943 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
10 | 8, 9 | sseqtrri 4001 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) |
11 | df-ss 3949 | . . . . 5 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)) | |
12 | 10, 11 | mpbi 231 | . . . 4 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴) |
13 | 3, 7, 12 | 3eqtr3g 2876 | . . 3 ⊢ (Fun 𝐹 → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) = (◡𝐹 “ 𝐴)) |
14 | 1, 2, 13 | 3syl 18 | . 2 ⊢ (𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) = (◡𝐹 “ 𝐴)) |
15 | i1frn 24205 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | |
16 | inss2 4203 | . . . 4 ⊢ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹 | |
17 | ssfi 8726 | . . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝐴 ∩ ran 𝐹) ∈ Fin) | |
18 | 15, 16, 17 | sylancl 586 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ∈ Fin) |
19 | i1fmbf 24203 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | |
20 | 19 | adantr 481 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹 ∈ MblFn) |
21 | 1 | adantr 481 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹:ℝ⟶ℝ) |
22 | 1 | frnd 6514 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ) |
23 | 16, 22 | sstrid 3975 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ⊆ ℝ) |
24 | 23 | sselda 3964 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝑦 ∈ ℝ) |
25 | mbfimasn 24160 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ 𝑦 ∈ ℝ) → (◡𝐹 “ {𝑦}) ∈ dom vol) | |
26 | 20, 21, 24, 25 | syl3anc 1363 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
27 | 26 | ralrimiva 3179 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
28 | finiunmbl 24072 | . . 3 ⊢ (((𝐴 ∩ ran 𝐹) ∈ Fin ∧ ∀𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) | |
29 | 18, 27, 28 | syl2anc 584 | . 2 ⊢ (𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
30 | 14, 29 | eqeltrrd 2911 | 1 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∩ cin 3932 ⊆ wss 3933 {csn 4557 ∪ ciun 4910 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 Fun wfun 6342 ⟶wf 6344 Fincfn 8497 ℝcr 10524 volcvol 23991 MblFncmbf 24142 ∫1citg1 24143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-pm 8398 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xadd 12496 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-xmet 20466 df-met 20467 df-ovol 23992 df-vol 23993 df-mbf 24147 df-itg1 24148 |
This theorem is referenced by: i1fima2 24207 itg1ge0 24214 i1fadd 24223 i1fmul 24224 itg1addlem2 24225 itg1addlem4 24227 itg1addlem5 24228 i1fmulc 24231 i1fres 24233 i1fpos 24234 itg1ge0a 24239 itg1climres 24242 itg2addnclem 34824 itg2addnclem2 34825 ftc1anclem3 34850 ftc1anclem6 34853 |
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