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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 25056 | . . 3 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
2 | ffun 6676 | . . 3 β’ (πΉ:ββΆβ β Fun πΉ) | |
3 | inpreima 7019 | . . . 4 β’ (Fun πΉ β (β‘πΉ β (π΄ β© ran πΉ)) = ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ))) | |
4 | iunid 5025 | . . . . . 6 β’ βͺ π¦ β (π΄ β© ran πΉ){π¦} = (π΄ β© ran πΉ) | |
5 | 4 | imaeq2i 6016 | . . . . 5 β’ (β‘πΉ β βͺ π¦ β (π΄ β© ran πΉ){π¦}) = (β‘πΉ β (π΄ β© ran πΉ)) |
6 | imaiun 7197 | . . . . 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 2767 | . . . 4 β’ (β‘πΉ β (π΄ β© ran πΉ)) = βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) |
8 | cnvimass 6038 | . . . . . 6 β’ (β‘πΉ β π΄) β dom πΉ | |
9 | cnvimarndm 6039 | . . . . . 6 β’ (β‘πΉ β ran πΉ) = dom πΉ | |
10 | 8, 9 | sseqtrri 3986 | . . . . 5 β’ (β‘πΉ β π΄) β (β‘πΉ β ran πΉ) |
11 | df-ss 3932 | . . . . 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 2800 | . . 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 25057 | . . . 4 β’ (πΉ β dom β«1 β ran πΉ β Fin) | |
16 | inss2 4194 | . . . 4 β’ (π΄ β© ran πΉ) β ran πΉ | |
17 | ssfi 9124 | . . . 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 587 | . . 3 β’ (πΉ β dom β«1 β (π΄ β© ran πΉ) β Fin) |
19 | i1fmbf 25055 | . . . . . 6 β’ (πΉ β dom β«1 β πΉ β MblFn) | |
20 | 19 | adantr 482 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β πΉ β MblFn) |
21 | 1 | adantr 482 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β πΉ:ββΆβ) |
22 | 1 | frnd 6681 | . . . . . . 7 β’ (πΉ β dom β«1 β ran πΉ β β) |
23 | 16, 22 | sstrid 3960 | . . . . . 6 β’ (πΉ β dom β«1 β (π΄ β© ran πΉ) β β) |
24 | 23 | sselda 3949 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β π¦ β β) |
25 | mbfimasn 25012 | . . . . 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 1372 | . . . 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 3144 | . . 3 β’ (πΉ β dom β«1 β βπ¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) β dom vol) |
28 | finiunmbl 24924 | . . 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 585 | . 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 2839 | 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 397 = wceq 1542 β wcel 2107 βwral 3065 β© cin 3914 β wss 3915 {csn 4591 βͺ ciun 4959 β‘ccnv 5637 dom cdm 5638 ran crn 5639 β cima 5641 Fun wfun 6495 βΆwf 6497 Fincfn 8890 βcr 11057 volcvol 24843 MblFncmbf 24994 β«1citg1 24995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-map 8774 df-pm 8775 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-n0 12421 df-z 12507 df-uz 12771 df-q 12881 df-rp 12923 df-xadd 13041 df-ioo 13275 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 df-xmet 20805 df-met 20806 df-ovol 24844 df-vol 24845 df-mbf 24999 df-itg1 25000 |
This theorem is referenced by: i1fima2 25059 itg1ge0 25066 i1fadd 25075 i1fmul 25076 itg1addlem2 25077 itg1addlem4 25079 itg1addlem4OLD 25080 itg1addlem5 25081 i1fmulc 25084 i1fres 25086 i1fpos 25087 itg1ge0a 25092 itg1climres 25095 itg2addnclem 36158 itg2addnclem2 36159 ftc1anclem3 36182 ftc1anclem6 36185 |
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