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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 2680 | . . . 4 ⊢ (𝐶 ∈ 𝐾 ↔ 𝐶 ∈ (𝐽 ↾t 𝐵)) |
3 | mbfimaopn.1 | . . . . . 6 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
4 | 3 | cnfldtop 22397 | . . . . 5 ⊢ 𝐽 ∈ Top |
5 | simp3 1056 | . . . . . 6 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → 𝐵 ⊆ ℂ) | |
6 | cnex 9896 | . . . . . 6 ⊢ ℂ ∈ V | |
7 | ssexg 4732 | . . . . . 6 ⊢ ((𝐵 ⊆ ℂ ∧ ℂ ∈ V) → 𝐵 ∈ V) | |
8 | 5, 6, 7 | sylancl 693 | . . . . 5 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → 𝐵 ∈ V) |
9 | elrest 15911 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝐵 ∈ V) → (𝐶 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵))) | |
10 | 4, 8, 9 | sylancr 694 | . . . 4 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (𝐶 ∈ (𝐽 ↾t 𝐵) ↔ ∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵))) |
11 | 2, 10 | syl5bb 271 | . . 3 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (𝐶 ∈ 𝐾 ↔ ∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵))) |
12 | simpl2 1058 | . . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → 𝐹:𝐴⟶𝐵) | |
13 | ffun 5961 | . . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → Fun 𝐹) | |
14 | inpreima 6250 | . . . . . . 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 23229 | . . . . . . . 8 ⊢ ((𝐹 ∈ MblFn ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝑢) ∈ dom vol) |
17 | 16 | 3ad2antl1 1216 | . . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝑢) ∈ dom vol) |
18 | fimacnv 6255 | . . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → (◡𝐹 “ 𝐵) = 𝐴) | |
19 | fdm 5964 | . . . . . . . . . 10 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴) | |
20 | 18, 19 | eqtr4d 2647 | . . . . . . . . 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 1057 | . . . . . . . . 9 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → 𝐹 ∈ MblFn) | |
23 | mbfdm 23201 | . . . . . . . . 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 2688 | . . . . . . 7 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ 𝐵) ∈ dom vol) |
26 | inmbl 23117 | . . . . . . 7 ⊢ (((◡𝐹 “ 𝑢) ∈ dom vol ∧ (◡𝐹 “ 𝐵) ∈ dom vol) → ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝐵)) ∈ dom vol) | |
27 | 17, 25, 26 | syl2anc 691 | . . . . . 6 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → ((◡𝐹 “ 𝑢) ∩ (◡𝐹 “ 𝐵)) ∈ dom vol) |
28 | 15, 27 | eqeltrd 2688 | . . . . 5 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (◡𝐹 “ (𝑢 ∩ 𝐵)) ∈ dom vol) |
29 | imaeq2 5381 | . . . . . 6 ⊢ (𝐶 = (𝑢 ∩ 𝐵) → (◡𝐹 “ 𝐶) = (◡𝐹 “ (𝑢 ∩ 𝐵))) | |
30 | 29 | eleq1d 2672 | . . . . 5 ⊢ (𝐶 = (𝑢 ∩ 𝐵) → ((◡𝐹 “ 𝐶) ∈ dom vol ↔ (◡𝐹 “ (𝑢 ∩ 𝐵)) ∈ dom vol)) |
31 | 28, 30 | syl5ibrcom 236 | . . . 4 ⊢ (((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) ∧ 𝑢 ∈ 𝐽) → (𝐶 = (𝑢 ∩ 𝐵) → (◡𝐹 “ 𝐶) ∈ dom vol)) |
32 | 31 | rexlimdva 3013 | . . 3 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (∃𝑢 ∈ 𝐽 𝐶 = (𝑢 ∩ 𝐵) → (◡𝐹 “ 𝐶) ∈ dom vol)) |
33 | 11, 32 | sylbid 229 | . 2 ⊢ ((𝐹 ∈ MblFn ∧ 𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ ℂ) → (𝐶 ∈ 𝐾 → (◡𝐹 “ 𝐶) ∈ dom vol)) |
34 | 33 | imp 444 | 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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ◡ccnv 5037 dom cdm 5038 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ↾t crest 15904 TopOpenctopn 15905 ℂfldccnfld 19567 Topctop 20517 volcvol 23039 MblFncmbf 23189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-ovol 23040 df-vol 23041 df-mbf 23194 |
This theorem is referenced by: cncombf 23231 |
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