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Mirrors > Home > MPE Home > Th. List > mbfid | Structured version Visualization version GIF version |
Description: The identity function is measurable. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
mbfid | β’ (π΄ β dom vol β ( I βΎ π΄) β MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvresima 6144 | . . . . 5 β’ (β‘( I βΎ π΄) β π₯) = ((β‘ I β π₯) β© π΄) | |
2 | cnvi 6056 | . . . . . . . 8 β’ β‘ I = I | |
3 | 2 | imaeq1i 5972 | . . . . . . 7 β’ (β‘ I β π₯) = ( I β π₯) |
4 | imai 5988 | . . . . . . 7 β’ ( I β π₯) = π₯ | |
5 | 3, 4 | eqtri 2764 | . . . . . 6 β’ (β‘ I β π₯) = π₯ |
6 | 5 | ineq1i 4148 | . . . . 5 β’ ((β‘ I β π₯) β© π΄) = (π₯ β© π΄) |
7 | 1, 6 | eqtri 2764 | . . . 4 β’ (β‘( I βΎ π΄) β π₯) = (π₯ β© π΄) |
8 | ioof 13221 | . . . . . . 7 β’ (,):(β* Γ β*)βΆπ« β | |
9 | ffn 6626 | . . . . . . 7 β’ ((,):(β* Γ β*)βΆπ« β β (,) Fn (β* Γ β*)) | |
10 | ovelrn 7476 | . . . . . . 7 β’ ((,) Fn (β* Γ β*) β (π₯ β ran (,) β βπ¦ β β* βπ§ β β* π₯ = (π¦(,)π§))) | |
11 | 8, 9, 10 | mp2b 10 | . . . . . 6 β’ (π₯ β ran (,) β βπ¦ β β* βπ§ β β* π₯ = (π¦(,)π§)) |
12 | id 22 | . . . . . . . . 9 β’ (π₯ = (π¦(,)π§) β π₯ = (π¦(,)π§)) | |
13 | ioombl 24770 | . . . . . . . . 9 β’ (π¦(,)π§) β dom vol | |
14 | 12, 13 | eqeltrdi 2845 | . . . . . . . 8 β’ (π₯ = (π¦(,)π§) β π₯ β dom vol) |
15 | 14 | a1i 11 | . . . . . . 7 β’ ((π¦ β β* β§ π§ β β*) β (π₯ = (π¦(,)π§) β π₯ β dom vol)) |
16 | 15 | rexlimivv 3193 | . . . . . 6 β’ (βπ¦ β β* βπ§ β β* π₯ = (π¦(,)π§) β π₯ β dom vol) |
17 | 11, 16 | sylbi 216 | . . . . 5 β’ (π₯ β ran (,) β π₯ β dom vol) |
18 | id 22 | . . . . 5 β’ (π΄ β dom vol β π΄ β dom vol) | |
19 | inmbl 24747 | . . . . 5 β’ ((π₯ β dom vol β§ π΄ β dom vol) β (π₯ β© π΄) β dom vol) | |
20 | 17, 18, 19 | syl2anr 598 | . . . 4 β’ ((π΄ β dom vol β§ π₯ β ran (,)) β (π₯ β© π΄) β dom vol) |
21 | 7, 20 | eqeltrid 2841 | . . 3 β’ ((π΄ β dom vol β§ π₯ β ran (,)) β (β‘( I βΎ π΄) β π₯) β dom vol) |
22 | 21 | ralrimiva 3140 | . 2 β’ (π΄ β dom vol β βπ₯ β ran (,)(β‘( I βΎ π΄) β π₯) β dom vol) |
23 | f1oi 6780 | . . . . 5 β’ ( I βΎ π΄):π΄β1-1-ontoβπ΄ | |
24 | f1of 6742 | . . . . 5 β’ (( I βΎ π΄):π΄β1-1-ontoβπ΄ β ( I βΎ π΄):π΄βΆπ΄) | |
25 | 23, 24 | ax-mp 5 | . . . 4 β’ ( I βΎ π΄):π΄βΆπ΄ |
26 | mblss 24736 | . . . 4 β’ (π΄ β dom vol β π΄ β β) | |
27 | fss 6643 | . . . 4 β’ ((( I βΎ π΄):π΄βΆπ΄ β§ π΄ β β) β ( I βΎ π΄):π΄βΆβ) | |
28 | 25, 26, 27 | sylancr 588 | . . 3 β’ (π΄ β dom vol β ( I βΎ π΄):π΄βΆβ) |
29 | ismbf 24833 | . . 3 β’ (( I βΎ π΄):π΄βΆβ β (( I βΎ π΄) β MblFn β βπ₯ β ran (,)(β‘( I βΎ π΄) β π₯) β dom vol)) | |
30 | 28, 29 | syl 17 | . 2 β’ (π΄ β dom vol β (( I βΎ π΄) β MblFn β βπ₯ β ran (,)(β‘( I βΎ π΄) β π₯) β dom vol)) |
31 | 22, 30 | mpbird 258 | 1 β’ (π΄ β dom vol β ( I βΎ π΄) β MblFn) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1539 β wcel 2104 βwral 3062 βwrex 3071 β© cin 3891 β wss 3892 π« cpw 4539 I cid 5495 Γ cxp 5594 β‘ccnv 5595 dom cdm 5596 ran crn 5597 βΎ cres 5598 β cima 5599 Fn wfn 6449 βΆwf 6450 β1-1-ontoβwf1o 6453 (class class class)co 7303 βcr 10912 β*cxr 11050 (,)cioo 13121 volcvol 24668 MblFncmbf 24819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 ax-inf2 9439 ax-cnex 10969 ax-resscn 10970 ax-1cn 10971 ax-icn 10972 ax-addcl 10973 ax-addrcl 10974 ax-mulcl 10975 ax-mulrcl 10976 ax-mulcom 10977 ax-addass 10978 ax-mulass 10979 ax-distr 10980 ax-i2m1 10981 ax-1ne0 10982 ax-1rid 10983 ax-rnegex 10984 ax-rrecex 10985 ax-cnre 10986 ax-pre-lttri 10987 ax-pre-lttrn 10988 ax-pre-ltadd 10989 ax-pre-mulgt0 10990 ax-pre-sup 10991 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3285 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-int 4887 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-se 5552 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-pred 6213 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-isom 6463 df-riota 7260 df-ov 7306 df-oprab 7307 df-mpo 7308 df-of 7561 df-om 7741 df-1st 7859 df-2nd 7860 df-frecs 8124 df-wrecs 8155 df-recs 8229 df-rdg 8268 df-1o 8324 df-2o 8325 df-er 8525 df-map 8644 df-pm 8645 df-en 8761 df-dom 8762 df-sdom 8763 df-fin 8764 df-sup 9241 df-inf 9242 df-oi 9309 df-dju 9699 df-card 9737 df-pnf 11053 df-mnf 11054 df-xr 11055 df-ltxr 11056 df-le 11057 df-sub 11249 df-neg 11250 df-div 11675 df-nn 12016 df-2 12078 df-3 12079 df-n0 12276 df-z 12362 df-uz 12625 df-q 12731 df-rp 12773 df-xadd 12891 df-ioo 13125 df-ico 13127 df-icc 13128 df-fz 13282 df-fzo 13425 df-fl 13554 df-seq 13764 df-exp 13825 df-hash 14087 df-cj 14851 df-re 14852 df-im 14853 df-sqrt 14987 df-abs 14988 df-clim 15238 df-rlim 15239 df-sum 15439 df-xmet 20631 df-met 20632 df-ovol 24669 df-vol 24670 df-mbf 24824 |
This theorem is referenced by: (None) |
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