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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 6249 | . . . . 5 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = ((◡ I “ 𝑥) ∩ 𝐴) | |
| 2 | cnvi 6160 | . . . . . . . 8 ⊢ ◡ I = I | |
| 3 | 2 | imaeq1i 6074 | . . . . . . 7 ⊢ (◡ I “ 𝑥) = ( I “ 𝑥) | 
| 4 | imai 6091 | . . . . . . 7 ⊢ ( I “ 𝑥) = 𝑥 | |
| 5 | 3, 4 | eqtri 2764 | . . . . . 6 ⊢ (◡ I “ 𝑥) = 𝑥 | 
| 6 | 5 | ineq1i 4215 | . . . . 5 ⊢ ((◡ I “ 𝑥) ∩ 𝐴) = (𝑥 ∩ 𝐴) | 
| 7 | 1, 6 | eqtri 2764 | . . . 4 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = (𝑥 ∩ 𝐴) | 
| 8 | ioof 13488 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
| 9 | ffn 6735 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
| 10 | ovelrn 7610 | . . . . . . 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 25601 | . . . . . . . . 9 ⊢ (𝑦(,)𝑧) ∈ dom vol | |
| 14 | 12, 13 | eqeltrdi 2848 | . . . . . . . 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 3200 | . . . . . 6 ⊢ (∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol) | 
| 17 | 11, 16 | sylbi 217 | . . . . 5 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ dom vol) | 
| 18 | id 22 | . . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ∈ dom vol) | |
| 19 | inmbl 25578 | . . . . 5 ⊢ ((𝑥 ∈ dom vol ∧ 𝐴 ∈ dom vol) → (𝑥 ∩ 𝐴) ∈ dom vol) | |
| 20 | 17, 18, 19 | syl2anr 597 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (𝑥 ∩ 𝐴) ∈ dom vol) | 
| 21 | 7, 20 | eqeltrid 2844 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) | 
| 22 | 21 | ralrimiva 3145 | . 2 ⊢ (𝐴 ∈ dom vol → ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) | 
| 23 | f1oi 6885 | . . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
| 24 | f1of 6847 | . . . . 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 25567 | . . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
| 27 | fss 6751 | . . . 4 ⊢ ((( I ↾ 𝐴):𝐴⟶𝐴 ∧ 𝐴 ⊆ ℝ) → ( I ↾ 𝐴):𝐴⟶ℝ) | |
| 28 | 25, 26, 27 | sylancr 587 | . . 3 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴):𝐴⟶ℝ) | 
| 29 | ismbf 25664 | . . 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 257 | 1 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 ∩ cin 3949 ⊆ wss 3950 𝒫 cpw 4599 I cid 5576 × cxp 5682 ◡ccnv 5683 dom cdm 5684 ran crn 5685 ↾ cres 5686 “ cima 5687 Fn wfn 6555 ⟶wf 6556 –1-1-onto→wf1o 6559 (class class class)co 7432 ℝcr 11155 ℝ*cxr 11295 (,)cioo 13388 volcvol 25499 MblFncmbf 25650 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-xadd 13156 df-ioo 13392 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-xmet 21358 df-met 21359 df-ovol 25500 df-vol 25501 df-mbf 25655 | 
| This theorem is referenced by: (None) | 
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