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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 24277 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
2 | ffun 6517 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹) | |
3 | inpreima 6834 | . . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
4 | iunid 4984 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦} = (𝐴 ∩ ran 𝐹) | |
5 | 4 | imaeq2i 5927 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)) |
6 | imaiun 7004 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) | |
7 | 5, 6 | eqtr3i 2846 | . . . 4 ⊢ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) |
8 | cnvimass 5949 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
9 | cnvimarndm 5950 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
10 | 8, 9 | sseqtrri 4004 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) |
11 | df-ss 3952 | . . . . 5 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)) | |
12 | 10, 11 | mpbi 232 | . . . 4 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴) |
13 | 3, 7, 12 | 3eqtr3g 2879 | . . 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 24278 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | |
16 | inss2 4206 | . . . 4 ⊢ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹 | |
17 | ssfi 8738 | . . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝐴 ∩ ran 𝐹) ∈ Fin) | |
18 | 15, 16, 17 | sylancl 588 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ∈ Fin) |
19 | i1fmbf 24276 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | |
20 | 19 | adantr 483 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹 ∈ MblFn) |
21 | 1 | adantr 483 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹:ℝ⟶ℝ) |
22 | 1 | frnd 6521 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ) |
23 | 16, 22 | sstrid 3978 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ⊆ ℝ) |
24 | 23 | sselda 3967 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝑦 ∈ ℝ) |
25 | mbfimasn 24233 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ 𝑦 ∈ ℝ) → (◡𝐹 “ {𝑦}) ∈ dom vol) | |
26 | 20, 21, 24, 25 | syl3anc 1367 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
27 | 26 | ralrimiva 3182 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
28 | finiunmbl 24145 | . . 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 586 | . 2 ⊢ (𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
30 | 14, 29 | eqeltrrd 2914 | 1 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 ⊆ wss 3936 {csn 4567 ∪ ciun 4919 ◡ccnv 5554 dom cdm 5555 ran crn 5556 “ cima 5558 Fun wfun 6349 ⟶wf 6351 Fincfn 8509 ℝcr 10536 volcvol 24064 MblFncmbf 24215 ∫1citg1 24216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-q 12350 df-rp 12391 df-xadd 12509 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-clim 14845 df-sum 15043 df-xmet 20538 df-met 20539 df-ovol 24065 df-vol 24066 df-mbf 24220 df-itg1 24221 |
This theorem is referenced by: i1fima2 24280 itg1ge0 24287 i1fadd 24296 i1fmul 24297 itg1addlem2 24298 itg1addlem4 24300 itg1addlem5 24301 i1fmulc 24304 i1fres 24306 i1fpos 24307 itg1ge0a 24312 itg1climres 24315 itg2addnclem 34958 itg2addnclem2 34959 ftc1anclem3 34984 ftc1anclem6 34987 |
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