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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 6230 | . . . . 5 β’ (β‘( I βΎ π΄) β π₯) = ((β‘ I β π₯) β© π΄) | |
2 | cnvi 6142 | . . . . . . . 8 β’ β‘ I = I | |
3 | 2 | imaeq1i 6057 | . . . . . . 7 β’ (β‘ I β π₯) = ( I β π₯) |
4 | imai 6074 | . . . . . . 7 β’ ( I β π₯) = π₯ | |
5 | 3, 4 | eqtri 2761 | . . . . . 6 β’ (β‘ I β π₯) = π₯ |
6 | 5 | ineq1i 4209 | . . . . 5 β’ ((β‘ I β π₯) β© π΄) = (π₯ β© π΄) |
7 | 1, 6 | eqtri 2761 | . . . 4 β’ (β‘( I βΎ π΄) β π₯) = (π₯ β© π΄) |
8 | ioof 13424 | . . . . . . 7 β’ (,):(β* Γ β*)βΆπ« β | |
9 | ffn 6718 | . . . . . . 7 β’ ((,):(β* Γ β*)βΆπ« β β (,) Fn (β* Γ β*)) | |
10 | ovelrn 7583 | . . . . . . 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 25082 | . . . . . . . . 9 β’ (π¦(,)π§) β dom vol | |
14 | 12, 13 | eqeltrdi 2842 | . . . . . . . 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 3200 | . . . . . 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 25059 | . . . . 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 2838 | . . 3 β’ ((π΄ β dom vol β§ π₯ β ran (,)) β (β‘( I βΎ π΄) β π₯) β dom vol) |
22 | 21 | ralrimiva 3147 | . 2 β’ (π΄ β dom vol β βπ₯ β ran (,)(β‘( I βΎ π΄) β π₯) β dom vol) |
23 | f1oi 6872 | . . . . 5 β’ ( I βΎ π΄):π΄β1-1-ontoβπ΄ | |
24 | f1of 6834 | . . . . 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 25048 | . . . 4 β’ (π΄ β dom vol β π΄ β β) | |
27 | fss 6735 | . . . 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 25145 | . . 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 257 | 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 1542 β wcel 2107 βwral 3062 βwrex 3071 β© cin 3948 β wss 3949 π« cpw 4603 I cid 5574 Γ cxp 5675 β‘ccnv 5676 dom cdm 5677 ran crn 5678 βΎ cres 5679 β cima 5680 Fn wfn 6539 βΆwf 6540 β1-1-ontoβwf1o 6543 (class class class)co 7409 βcr 11109 β*cxr 11247 (,)cioo 13324 volcvol 24980 MblFncmbf 25131 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xadd 13093 df-ioo 13328 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433 df-sum 15633 df-xmet 20937 df-met 20938 df-ovol 24981 df-vol 24982 df-mbf 25136 |
This theorem is referenced by: (None) |
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