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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 25604 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → 𝐹:ℝ⟶ℝ) | |
2 | ffun 6725 | . . 3 ⊢ (𝐹:ℝ⟶ℝ → Fun 𝐹) | |
3 | inpreima 7073 | . . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹))) | |
4 | iunid 5063 | . . . . . 6 ⊢ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦} = (𝐴 ∩ ran 𝐹) | |
5 | 4 | imaeq2i 6061 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = (◡𝐹 “ (𝐴 ∩ ran 𝐹)) |
6 | imaiun 7255 | . . . . 5 ⊢ (◡𝐹 “ ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹){𝑦}) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) | |
7 | 5, 6 | eqtr3i 2758 | . . . 4 ⊢ (◡𝐹 “ (𝐴 ∩ ran 𝐹)) = ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) |
8 | cnvimass 6085 | . . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹 | |
9 | cnvimarndm 6086 | . . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹 | |
10 | 8, 9 | sseqtrri 4017 | . . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) |
11 | df-ss 3964 | . . . . 5 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)) | |
12 | 10, 11 | mpbi 229 | . . . 4 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴) |
13 | 3, 7, 12 | 3eqtr3g 2791 | . . 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 25605 | . . . 4 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin) | |
16 | inss2 4230 | . . . 4 ⊢ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹 | |
17 | ssfi 9197 | . . . 4 ⊢ ((ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) ⊆ ran 𝐹) → (𝐴 ∩ ran 𝐹) ∈ Fin) | |
18 | 15, 16, 17 | sylancl 585 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ∈ Fin) |
19 | i1fmbf 25603 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → 𝐹 ∈ MblFn) | |
20 | 19 | adantr 480 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹 ∈ MblFn) |
21 | 1 | adantr 480 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝐹:ℝ⟶ℝ) |
22 | 1 | frnd 6730 | . . . . . . 7 ⊢ (𝐹 ∈ dom ∫1 → ran 𝐹 ⊆ ℝ) |
23 | 16, 22 | sstrid 3991 | . . . . . 6 ⊢ (𝐹 ∈ dom ∫1 → (𝐴 ∩ ran 𝐹) ⊆ ℝ) |
24 | 23 | sselda 3980 | . . . . 5 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → 𝑦 ∈ ℝ) |
25 | mbfimasn 25560 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ 𝑦 ∈ ℝ) → (◡𝐹 “ {𝑦}) ∈ dom vol) | |
26 | 20, 21, 24, 25 | syl3anc 1369 | . . . 4 ⊢ ((𝐹 ∈ dom ∫1 ∧ 𝑦 ∈ (𝐴 ∩ ran 𝐹)) → (◡𝐹 “ {𝑦}) ∈ dom vol) |
27 | 26 | ralrimiva 3143 | . . 3 ⊢ (𝐹 ∈ dom ∫1 → ∀𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
28 | finiunmbl 25472 | . . 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 583 | . 2 ⊢ (𝐹 ∈ dom ∫1 → ∪ 𝑦 ∈ (𝐴 ∩ ran 𝐹)(◡𝐹 “ {𝑦}) ∈ dom vol) |
30 | 14, 29 | eqeltrrd 2830 | 1 ⊢ (𝐹 ∈ dom ∫1 → (◡𝐹 “ 𝐴) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ∩ cin 3946 ⊆ wss 3947 {csn 4629 ∪ ciun 4996 ◡ccnv 5677 dom cdm 5678 ran crn 5679 “ cima 5681 Fun wfun 6542 ⟶wf 6544 Fincfn 8963 ℝcr 11137 volcvol 25391 MblFncmbf 25542 ∫1citg1 25543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-map 8846 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-q 12963 df-rp 13007 df-xadd 13125 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-sum 15665 df-xmet 21271 df-met 21272 df-ovol 25392 df-vol 25393 df-mbf 25547 df-itg1 25548 |
This theorem is referenced by: i1fima2 25607 itg1ge0 25614 i1fadd 25623 i1fmul 25624 itg1addlem2 25625 itg1addlem4 25627 itg1addlem4OLD 25628 itg1addlem5 25629 i1fmulc 25632 i1fres 25634 i1fpos 25635 itg1ge0a 25640 itg1climres 25643 itg2addnclem 37144 itg2addnclem2 37145 ftc1anclem3 37168 ftc1anclem6 37171 |
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