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Mirrors > Home > MPE Home > Th. List > mbfimaopn2 | Structured version Visualization version GIF version |
Description: The preimage of any set open in the subspace topology of the range of the function is measurable. (Contributed by Mario Carneiro, 25-Aug-2014.) |
Ref | Expression |
---|---|
mbfimaopn.1 | ⊢ 𝐽 = (TopOpen‘ℂfld) |
mbfimaopn2.2 | ⊢ 𝐾 = (𝐽 ↾t 𝐵) |
Ref | Expression |
---|---|
mbfimaopn2 | ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝐶 ∈ 𝐾) → (◡𝐹 “ 𝐶) ∈ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfimaopn2.2 | . . . . 5 ⊢ 𝐾 = (𝐽 ↾t 𝐵) | |
2 | 1 | eleq2i 2693 | . . . 4 ⊢ (𝐶 ∈ 𝐾 ↔ 𝐶 ∈ (𝐽 ↾t 𝐵)) |
3 | mbfimaopn.1 | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
4 | 3 | cnfldtop 22587 | . . . . 5 ⊢ 𝐽 ∈ Top |
5 | simp3 1063 | . . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → 𝐵 ⊆ ℂ) | |
6 | cnex 10017 | . . . . . 6 ⊢ ℂ ∈ V | |
7 | ssexg 4804 | . . . . . 6 ⊢ ((𝐵 ⊆ ℂ ∧ ℂ ∈ V) → 𝐵 ∈ V) | |
8 | 5, 6, 7 | sylancl 694 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → 𝐵 ∈ V) |
9 | elrest 16088 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ V) → (𝐶 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵))) | |
10 | 4, 8, 9 | sylancr 695 | . . . 4 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (𝐶 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵))) |
11 | 2, 10 | syl5bb 272 | . . 3 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (𝐶 ∈ 𝐾 ↔ ∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵))) |
12 | simpl2 1065 | . . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → 𝐹:𝐴⟶𝐵) | |
13 | ffun 6048 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
14 | inpreima 6342 | . . . . . . 7 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝑢 ∩ 𝐵)) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝐵))) | |
15 | 12, 13, 14 | 3syl 18 | . . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ (𝑢 ∩ 𝐵)) = ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝐵))) |
16 | 3 | mbfimaopn 23423 | . . . . . . . 8 ⊢ ((𝐹 ∈ MblFn ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝑢) ∈ dom vol) |
17 | 16 | 3ad2antl1 1223 | . . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝑢) ∈ dom vol) |
18 | fimacnv 6347 | . . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) | |
19 | fdm 6051 | . . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
20 | 18, 19 | eqtr4d 2659 | . . . . . . . . 9 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = dom 𝐹) |
21 | 12, 20 | syl 17 | . . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝐵) = dom 𝐹) |
22 | simpl1 1064 | . . . . . . . . 9 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → 𝐹 ∈ MblFn) | |
23 | mbfdm 23395 | . . . . . . . . 9 ⊢ (𝐹 ∈ MblFn → dom 𝐹 ∈ dom vol) | |
24 | 22, 23 | syl 17 | . . . . . . . 8 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → dom 𝐹 ∈ dom vol) |
25 | 21, 24 | eqeltrd 2701 | . . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝐵) ∈ dom vol) |
26 | inmbl 23310 | . . . . . . 7 ⊢ (((◡𝐹 “ 𝑢) ∈ dom vol ∧ (◡𝐹 “ 𝐵) ∈ dom vol) → ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝐵)) ∈ dom vol) | |
27 | 17, 25, 26 | syl2anc 693 | . . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝐵)) ∈ dom vol) |
28 | 15, 27 | eqeltrd 2701 | . . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ (𝑢 ∩ 𝐵)) ∈ dom vol) |
29 | imaeq2 5462 | . . . . . 6 ⊢ (𝐶 = (𝑢 ∩ 𝐵) → (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝑢 ∩ 𝐵))) | |
30 | 29 | eleq1d 2686 | . . . . 5 ⊢ (𝐶 = (𝑢 ∩ 𝐵) → ((◡𝐹 “ 𝐶) ∈ dom vol ↔ (◡𝐹 “ (𝑢 ∩ 𝐵)) ∈ dom vol)) |
31 | 28, 30 | syl5ibrcom 237 | . . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (𝐶 = (𝑢 ∩ 𝐵) → (◡𝐹 “ 𝐶) ∈ dom vol)) |
32 | 31 | rexlimdva 3031 | . . 3 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵) → (◡𝐹 “ 𝐶) ∈ dom vol)) |
33 | 11, 32 | sylbid 230 | . 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (𝐶 ∈ 𝐾 → (◡𝐹 “ 𝐶) ∈ dom vol)) |
34 | 33 | imp 445 | 1 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝐶 ∈ 𝐾) → (◡𝐹 “ 𝐶) ∈ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 ◡ccnv 5113 dom cdm 5114 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ↾t crest 16081 TopOpenctopn 16082 ℂfldccnfld 19746 Topctop 20698 volcvol 23232 MblFncmbf 23383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 |
This theorem is referenced by: cncombf 23425 |
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