![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 23880 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
2 | ffun 6294 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹) | |
3 | inpreima 6606 | . . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
4 | iunid 4808 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦} = (𝐴 ∩ ran 𝐹) | |
5 | 4 | imaeq2i 5718 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)) |
6 | imaiun 6775 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) | |
7 | 5, 6 | eqtr3i 2804 | . . . 4 ⊢ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) |
8 | cnvimass 5739 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
9 | cnvimarndm 5740 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
10 | 8, 9 | sseqtr4i 3857 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) |
11 | df-ss 3806 | . . . . 5 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)) | |
12 | 10, 11 | mpbi 222 | . . . 4 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴) |
13 | 3, 7, 12 | 3eqtr3g 2837 | . . 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 23881 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | |
16 | inss2 4054 | . . . 4 ⊢ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹 | |
17 | ssfi 8468 | . . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝐴 ∩ ran 𝐹) ∈ Fin) | |
18 | 15, 16, 17 | sylancl 580 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ∈ Fin) |
19 | i1fmbf 23879 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | |
20 | 19 | adantr 474 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹 ∈ MblFn) |
21 | 1 | adantr 474 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹:ℝ⟶ℝ) |
22 | 1 | frnd 6298 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ) |
23 | 16, 22 | syl5ss 3832 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ⊆ ℝ) |
24 | 23 | sselda 3821 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝑦 ∈ ℝ) |
25 | mbfimasn 23836 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ 𝑦 ∈ ℝ) → (◡𝐹 “ {𝑦}) ∈ dom vol) | |
26 | 20, 21, 24, 25 | syl3anc 1439 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
27 | 26 | ralrimiva 3148 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
28 | finiunmbl 23748 | . . 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 579 | . 2 ⊢ (𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
30 | 14, 29 | eqeltrrd 2860 | 1 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∩ cin 3791 ⊆ wss 3792 {csn 4398 ∪ ciun 4753 ◡ccnv 5354 dom cdm 5355 ran crn 5356 “ cima 5358 Fun wfun 6129 ⟶wf 6131 Fincfn 8241 ℝcr 10271 volcvol 23667 MblFncmbf 23818 ∫1citg1 23819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-q 12096 df-rp 12138 df-xadd 12258 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-xmet 20135 df-met 20136 df-ovol 23668 df-vol 23669 df-mbf 23823 df-itg1 23824 |
This theorem is referenced by: i1fima2 23883 itg1ge0 23890 i1fadd 23899 i1fmul 23900 itg1addlem2 23901 itg1addlem4 23903 itg1addlem5 23904 i1fmulc 23907 i1fres 23909 i1fpos 23910 itg1ge0a 23915 itg1climres 23918 itg2addnclem 34086 itg2addnclem2 34087 ftc1anclem3 34112 ftc1anclem6 34115 |
Copyright terms: Public domain | W3C validator |