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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 | ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvresima 6089 | . . . . 5 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = ((◡ I “ 𝑥) ∩ 𝐴) | |
2 | cnvi 6002 | . . . . . . . 8 ⊢ ◡ I = I | |
3 | 2 | imaeq1i 5928 | . . . . . . 7 ⊢ (◡ I “ 𝑥) = ( I “ 𝑥) |
4 | imai 5944 | . . . . . . 7 ⊢ ( I “ 𝑥) = 𝑥 | |
5 | 3, 4 | eqtri 2846 | . . . . . 6 ⊢ (◡ I “ 𝑥) = 𝑥 |
6 | 5 | ineq1i 4187 | . . . . 5 ⊢ ((◡ I “ 𝑥) ∩ 𝐴) = (𝑥 ∩ 𝐴) |
7 | 1, 6 | eqtri 2846 | . . . 4 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = (𝑥 ∩ 𝐴) |
8 | ioof 12838 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
9 | ffn 6516 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
10 | ovelrn 7326 | . . . . . . 7 ⊢ ((,) Fn (ℝ* × ℝ*) → (𝑥 ∈ ran (,) ↔ ∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧))) | |
11 | 8, 9, 10 | mp2b 10 | . . . . . 6 ⊢ (𝑥 ∈ ran (,) ↔ ∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧)) |
12 | id 22 | . . . . . . . . 9 ⊢ (𝑥 = (𝑦(,)𝑧) → 𝑥 = (𝑦(,)𝑧)) | |
13 | ioombl 24168 | . . . . . . . . 9 ⊢ (𝑦(,)𝑧) ∈ dom vol | |
14 | 12, 13 | eqeltrdi 2923 | . . . . . . . 8 ⊢ (𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol) |
15 | 14 | a1i 11 | . . . . . . 7 ⊢ ((𝑦 ∈ ℝ* ∧ 𝑧 ∈ ℝ*) → (𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol)) |
16 | 15 | rexlimivv 3294 | . . . . . 6 ⊢ (∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol) |
17 | 11, 16 | sylbi 219 | . . . . 5 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ dom vol) |
18 | id 22 | . . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ∈ dom vol) | |
19 | inmbl 24145 | . . . . 5 ⊢ ((𝑥 ∈ dom vol ∧ 𝐴 ∈ dom vol) → (𝑥 ∩ 𝐴) ∈ dom vol) | |
20 | 17, 18, 19 | syl2anr 598 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (𝑥 ∩ 𝐴) ∈ dom vol) |
21 | 7, 20 | eqeltrid 2919 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
22 | 21 | ralrimiva 3184 | . 2 ⊢ (𝐴 ∈ dom vol → ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
23 | f1oi 6654 | . . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
24 | f1of 6617 | . . . . 5 ⊢ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 → ( I ↾ 𝐴):𝐴⟶𝐴) | |
25 | 23, 24 | ax-mp 5 | . . . 4 ⊢ ( I ↾ 𝐴):𝐴⟶𝐴 |
26 | mblss 24134 | . . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
27 | fss 6529 | . . . 4 ⊢ ((( I ↾ 𝐴):𝐴⟶𝐴 ∧ 𝐴 ⊆ ℝ) → ( I ↾ 𝐴):𝐴⟶ℝ) | |
28 | 25, 26, 27 | sylancr 589 | . . 3 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴):𝐴⟶ℝ) |
29 | ismbf 24231 | . . 3 ⊢ (( I ↾ 𝐴):𝐴⟶ℝ → (( I ↾ 𝐴) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol)) | |
30 | 28, 29 | syl 17 | . 2 ⊢ (𝐴 ∈ dom vol → (( I ↾ 𝐴) ∈ MblFn ↔ ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol)) |
31 | 22, 30 | mpbird 259 | 1 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 ∩ cin 3937 ⊆ wss 3938 𝒫 cpw 4541 I cid 5461 × cxp 5555 ◡ccnv 5556 dom cdm 5557 ran crn 5558 ↾ cres 5559 “ cima 5560 Fn wfn 6352 ⟶wf 6353 –1-1-onto→wf1o 6356 (class class class)co 7158 ℝcr 10538 ℝ*cxr 10676 (,)cioo 12741 volcvol 24066 MblFncmbf 24217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-dju 9332 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xadd 12511 df-ioo 12745 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-xmet 20540 df-met 20541 df-ovol 24067 df-vol 24068 df-mbf 24222 |
This theorem is referenced by: (None) |
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