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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 | β’ π½ = (TopOpenββfld) |
| mbfimaopn2.2 | β’ πΎ = (π½ βΎt π΅) |
| Ref | Expression |
|---|---|
| mbfimaopn2 | β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ πΆ β πΎ) β (β‘πΉ β πΆ) β dom vol) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mbfimaopn2.2 | . . . . 5 β’ πΎ = (π½ βΎt π΅) | |
| 2 | 1 | eleq2i 2824 | . . . 4 β’ (πΆ β πΎ β πΆ β (π½ βΎt π΅)) |
| 3 | mbfimaopn.1 | . . . . . 6 β’ π½ = (TopOpenββfld) | |
| 4 | 3 | cnfldtop 24521 | . . . . 5 β’ π½ β Top |
| 5 | simp3 1137 | . . . . . 6 β’ ((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β π΅ β β) | |
| 6 | cnex 11195 | . . . . . 6 β’ β β V | |
| 7 | ssexg 5323 | . . . . . 6 β’ ((π΅ β β β§ β β V) β π΅ β V) | |
| 8 | 5, 6, 7 | sylancl 585 | . . . . 5 β’ ((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β π΅ β V) |
| 9 | elrest 17378 | . . . . 5 β’ ((π½ β Top β§ π΅ β V) β (πΆ β (π½ βΎt π΅) β βπ’ β π½ πΆ = (π’ β© π΅))) | |
| 10 | 4, 8, 9 | sylancr 586 | . . . 4 β’ ((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β (πΆ β (π½ βΎt π΅) β βπ’ β π½ πΆ = (π’ β© π΅))) |
| 11 | 2, 10 | bitrid 283 | . . 3 β’ ((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β (πΆ β πΎ β βπ’ β π½ πΆ = (π’ β© π΅))) |
| 12 | simpl2 1191 | . . . . . . 7 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β πΉ:π΄βΆπ΅) | |
| 13 | ffun 6720 | . . . . . . 7 β’ (πΉ:π΄βΆπ΅ β Fun πΉ) | |
| 14 | inpreima 7065 | . . . . . . 7 β’ (Fun πΉ β (β‘πΉ β (π’ β© π΅)) = ((β‘πΉ β π’) β© (β‘πΉ β π΅))) | |
| 15 | 12, 13, 14 | 3syl 18 | . . . . . 6 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β (β‘πΉ β (π’ β© π΅)) = ((β‘πΉ β π’) β© (β‘πΉ β π΅))) |
| 16 | 3 | mbfimaopn 25406 | . . . . . . . 8 β’ ((πΉ β MblFn β§ π’ β π½) β (β‘πΉ β π’) β dom vol) |
| 17 | 16 | 3ad2antl1 1184 | . . . . . . 7 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β (β‘πΉ β π’) β dom vol) |
| 18 | fimacnv 6739 | . . . . . . . . . 10 β’ (πΉ:π΄βΆπ΅ β (β‘πΉ β π΅) = π΄) | |
| 19 | fdm 6726 | . . . . . . . . . 10 β’ (πΉ:π΄βΆπ΅ β dom πΉ = π΄) | |
| 20 | 18, 19 | eqtr4d 2774 | . . . . . . . . 9 β’ (πΉ:π΄βΆπ΅ β (β‘πΉ β π΅) = dom πΉ) |
| 21 | 12, 20 | syl 17 | . . . . . . . 8 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β (β‘πΉ β π΅) = dom πΉ) |
| 22 | simpl1 1190 | . . . . . . . . 9 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β πΉ β MblFn) | |
| 23 | mbfdm 25376 | . . . . . . . . 9 β’ (πΉ β MblFn β dom πΉ β dom vol) | |
| 24 | 22, 23 | syl 17 | . . . . . . . 8 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β dom πΉ β dom vol) |
| 25 | 21, 24 | eqeltrd 2832 | . . . . . . 7 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β (β‘πΉ β π΅) β dom vol) |
| 26 | inmbl 25292 | . . . . . . 7 β’ (((β‘πΉ β π’) β dom vol β§ (β‘πΉ β π΅) β dom vol) β ((β‘πΉ β π’) β© (β‘πΉ β π΅)) β dom vol) | |
| 27 | 17, 25, 26 | syl2anc 583 | . . . . . 6 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β ((β‘πΉ β π’) β© (β‘πΉ β π΅)) β dom vol) |
| 28 | 15, 27 | eqeltrd 2832 | . . . . 5 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β (β‘πΉ β (π’ β© π΅)) β dom vol) |
| 29 | imaeq2 6055 | . . . . . 6 β’ (πΆ = (π’ β© π΅) β (β‘πΉ β πΆ) = (β‘πΉ β (π’ β© π΅))) | |
| 30 | 29 | eleq1d 2817 | . . . . 5 β’ (πΆ = (π’ β© π΅) β ((β‘πΉ β πΆ) β dom vol β (β‘πΉ β (π’ β© π΅)) β dom vol)) |
| 31 | 28, 30 | syl5ibrcom 246 | . . . 4 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ π’ β π½) β (πΆ = (π’ β© π΅) β (β‘πΉ β πΆ) β dom vol)) |
| 32 | 31 | rexlimdva 3154 | . . 3 β’ ((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β (βπ’ β π½ πΆ = (π’ β© π΅) β (β‘πΉ β πΆ) β dom vol)) |
| 33 | 11, 32 | sylbid 239 | . 2 β’ ((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β (πΆ β πΎ β (β‘πΉ β πΆ) β dom vol)) |
| 34 | 33 | imp 406 | 1 β’ (((πΉ β MblFn β§ πΉ:π΄βΆπ΅ β§ π΅ β β) β§ πΆ β πΎ) β (β‘πΉ β πΆ) β dom vol) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwrex 3069 Vcvv 3473 β© cin 3947 β wss 3948 β‘ccnv 5675 dom cdm 5676 β cima 5679 Fun wfun 6537 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcc 11112 βΎt crest 17371 TopOpenctopn 17372 βfldccnfld 21145 Topctop 22616 volcvol 25213 MblFncmbf 25364 |
| 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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-acn 9941 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-sum 15638 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cn 22952 df-cnp 22953 df-tx 23287 df-hmeo 23480 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-ovol 25214 df-vol 25215 df-mbf 25369 |
| This theorem is referenced by: cncombf 25408 |
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