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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 | β’ (πΉ β dom β«1 β (β‘πΉ β π΄) β dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | i1ff 25556 | . . 3 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
2 | ffun 6713 | . . 3 β’ (πΉ:ββΆβ β Fun πΉ) | |
3 | inpreima 7058 | . . . 4 β’ (Fun πΉ β (β‘πΉ β (π΄ β© ran πΉ)) = ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ))) | |
4 | iunid 5056 | . . . . . 6 β’ βͺ π¦ β (π΄ β© ran πΉ){π¦} = (π΄ β© ran πΉ) | |
5 | 4 | imaeq2i 6050 | . . . . 5 β’ (β‘πΉ β βͺ π¦ β (π΄ β© ran πΉ){π¦}) = (β‘πΉ β (π΄ β© ran πΉ)) |
6 | imaiun 7239 | . . . . 5 β’ (β‘πΉ β βͺ π¦ β (π΄ β© ran πΉ){π¦}) = βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) | |
7 | 5, 6 | eqtr3i 2756 | . . . 4 β’ (β‘πΉ β (π΄ β© ran πΉ)) = βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) |
8 | cnvimass 6073 | . . . . . 6 β’ (β‘πΉ β π΄) β dom πΉ | |
9 | cnvimarndm 6074 | . . . . . 6 β’ (β‘πΉ β ran πΉ) = dom πΉ | |
10 | 8, 9 | sseqtrri 4014 | . . . . 5 β’ (β‘πΉ β π΄) β (β‘πΉ β ran πΉ) |
11 | df-ss 3960 | . . . . 5 β’ ((β‘πΉ β π΄) β (β‘πΉ β ran πΉ) β ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ)) = (β‘πΉ β π΄)) | |
12 | 10, 11 | mpbi 229 | . . . 4 β’ ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ)) = (β‘πΉ β π΄) |
13 | 3, 7, 12 | 3eqtr3g 2789 | . . 3 β’ (Fun πΉ β βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) = (β‘πΉ β π΄)) |
14 | 1, 2, 13 | 3syl 18 | . 2 β’ (πΉ β dom β«1 β βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) = (β‘πΉ β π΄)) |
15 | i1frn 25557 | . . . 4 β’ (πΉ β dom β«1 β ran πΉ β Fin) | |
16 | inss2 4224 | . . . 4 β’ (π΄ β© ran πΉ) β ran πΉ | |
17 | ssfi 9172 | . . . 4 β’ ((ran πΉ β Fin β§ (π΄ β© ran πΉ) β ran πΉ) β (π΄ β© ran πΉ) β Fin) | |
18 | 15, 16, 17 | sylancl 585 | . . 3 β’ (πΉ β dom β«1 β (π΄ β© ran πΉ) β Fin) |
19 | i1fmbf 25555 | . . . . . 6 β’ (πΉ β dom β«1 β πΉ β MblFn) | |
20 | 19 | adantr 480 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β πΉ β MblFn) |
21 | 1 | adantr 480 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β πΉ:ββΆβ) |
22 | 1 | frnd 6718 | . . . . . . 7 β’ (πΉ β dom β«1 β ran πΉ β β) |
23 | 16, 22 | sstrid 3988 | . . . . . 6 β’ (πΉ β dom β«1 β (π΄ β© ran πΉ) β β) |
24 | 23 | sselda 3977 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β π¦ β β) |
25 | mbfimasn 25512 | . . . . 5 β’ ((πΉ β MblFn β§ πΉ:ββΆβ β§ π¦ β β) β (β‘πΉ β {π¦}) β dom vol) | |
26 | 20, 21, 24, 25 | syl3anc 1368 | . . . 4 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β (β‘πΉ β {π¦}) β dom vol) |
27 | 26 | ralrimiva 3140 | . . 3 β’ (πΉ β dom β«1 β βπ¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) β dom vol) |
28 | finiunmbl 25424 | . . 3 β’ (((π΄ β© ran πΉ) β Fin β§ βπ¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) β dom vol) β βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) β dom vol) | |
29 | 18, 27, 28 | syl2anc 583 | . 2 β’ (πΉ β dom β«1 β βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) β dom vol) |
30 | 14, 29 | eqeltrrd 2828 | 1 β’ (πΉ β dom β«1 β (β‘πΉ β π΄) β dom vol) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βwral 3055 β© cin 3942 β wss 3943 {csn 4623 βͺ ciun 4990 β‘ccnv 5668 dom cdm 5669 ran crn 5670 β cima 5672 Fun wfun 6530 βΆwf 6532 Fincfn 8938 βcr 11108 volcvol 25343 MblFncmbf 25494 β«1citg1 25495 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-seq 13970 df-exp 14031 df-hash 14294 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-sum 15637 df-xmet 21229 df-met 21230 df-ovol 25344 df-vol 25345 df-mbf 25499 df-itg1 25500 |
This theorem is referenced by: i1fima2 25559 itg1ge0 25566 i1fadd 25575 i1fmul 25576 itg1addlem2 25577 itg1addlem4 25579 itg1addlem4OLD 25580 itg1addlem5 25581 i1fmulc 25584 i1fres 25586 i1fpos 25587 itg1ge0a 25592 itg1climres 25595 itg2addnclem 37050 itg2addnclem2 37051 ftc1anclem3 37074 ftc1anclem6 37077 |
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