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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 25625 | . . 3 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
2 | ffun 6730 | . . 3 β’ (πΉ:ββΆβ β Fun πΉ) | |
3 | inpreima 7078 | . . . 4 β’ (Fun πΉ β (β‘πΉ β (π΄ β© ran πΉ)) = ((β‘πΉ β π΄) β© (β‘πΉ β ran πΉ))) | |
4 | iunid 5067 | . . . . . 6 β’ βͺ π¦ β (π΄ β© ran πΉ){π¦} = (π΄ β© ran πΉ) | |
5 | 4 | imaeq2i 6066 | . . . . 5 β’ (β‘πΉ β βͺ π¦ β (π΄ β© ran πΉ){π¦}) = (β‘πΉ β (π΄ β© ran πΉ)) |
6 | imaiun 7261 | . . . . 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 2758 | . . . 4 β’ (β‘πΉ β (π΄ β© ran πΉ)) = βͺ π¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) |
8 | cnvimass 6090 | . . . . . 6 β’ (β‘πΉ β π΄) β dom πΉ | |
9 | cnvimarndm 6091 | . . . . . 6 β’ (β‘πΉ β ran πΉ) = dom πΉ | |
10 | 8, 9 | sseqtrri 4019 | . . . . 5 β’ (β‘πΉ β π΄) β (β‘πΉ β ran πΉ) |
11 | df-ss 3966 | . . . . 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 2791 | . . 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 25626 | . . . 4 β’ (πΉ β dom β«1 β ran πΉ β Fin) | |
16 | inss2 4232 | . . . 4 β’ (π΄ β© ran πΉ) β ran πΉ | |
17 | ssfi 9204 | . . . 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 584 | . . 3 β’ (πΉ β dom β«1 β (π΄ β© ran πΉ) β Fin) |
19 | i1fmbf 25624 | . . . . . 6 β’ (πΉ β dom β«1 β πΉ β MblFn) | |
20 | 19 | adantr 479 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β πΉ β MblFn) |
21 | 1 | adantr 479 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β πΉ:ββΆβ) |
22 | 1 | frnd 6735 | . . . . . . 7 β’ (πΉ β dom β«1 β ran πΉ β β) |
23 | 16, 22 | sstrid 3993 | . . . . . 6 β’ (πΉ β dom β«1 β (π΄ β© ran πΉ) β β) |
24 | 23 | sselda 3982 | . . . . 5 β’ ((πΉ β dom β«1 β§ π¦ β (π΄ β© ran πΉ)) β π¦ β β) |
25 | mbfimasn 25581 | . . . . 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 3143 | . . 3 β’ (πΉ β dom β«1 β βπ¦ β (π΄ β© ran πΉ)(β‘πΉ β {π¦}) β dom vol) |
28 | finiunmbl 25493 | . . 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 582 | . 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 2830 | 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 394 = wceq 1533 β wcel 2098 βwral 3058 β© cin 3948 β wss 3949 {csn 4632 βͺ ciun 5000 β‘ccnv 5681 dom cdm 5682 ran crn 5683 β cima 5685 Fun wfun 6547 βΆwf 6549 Fincfn 8970 βcr 11145 volcvol 25412 MblFncmbf 25563 β«1citg1 25564 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 df-mbf 25568 df-itg1 25569 |
This theorem is referenced by: i1fima2 25628 itg1ge0 25635 i1fadd 25644 i1fmul 25645 itg1addlem2 25646 itg1addlem4 25648 itg1addlem4OLD 25649 itg1addlem5 25650 i1fmulc 25653 i1fres 25655 i1fpos 25656 itg1ge0a 25661 itg1climres 25664 itg2addnclem 37177 itg2addnclem2 37178 ftc1anclem3 37201 ftc1anclem6 37204 |
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