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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 6081 | . . . . 5 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = ((◡ I “ 𝑥) ∩ 𝐴) | |
2 | cnvi 5994 | . . . . . . . 8 ⊢ ◡ I = I | |
3 | 2 | imaeq1i 5920 | . . . . . . 7 ⊢ (◡ I “ 𝑥) = ( I “ 𝑥) |
4 | imai 5936 | . . . . . . 7 ⊢ ( I “ 𝑥) = 𝑥 | |
5 | 3, 4 | eqtri 2844 | . . . . . 6 ⊢ (◡ I “ 𝑥) = 𝑥 |
6 | 5 | ineq1i 4184 | . . . . 5 ⊢ ((◡ I “ 𝑥) ∩ 𝐴) = (𝑥 ∩ 𝐴) |
7 | 1, 6 | eqtri 2844 | . . . 4 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = (𝑥 ∩ 𝐴) |
8 | ioof 12825 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
9 | ffn 6508 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
10 | ovelrn 7313 | . . . . . . 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 24095 | . . . . . . . . 9 ⊢ (𝑦(,)𝑧) ∈ dom vol | |
14 | 12, 13 | syl6eqel 2921 | . . . . . . . 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 3292 | . . . . . 6 ⊢ (∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol) |
17 | 11, 16 | sylbi 218 | . . . . 5 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ dom vol) |
18 | id 22 | . . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ∈ dom vol) | |
19 | inmbl 24072 | . . . . 5 ⊢ ((𝑥 ∈ dom vol ∧ 𝐴 ∈ dom vol) → (𝑥 ∩ 𝐴) ∈ dom vol) | |
20 | 17, 18, 19 | syl2anr 596 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (𝑥 ∩ 𝐴) ∈ dom vol) |
21 | 7, 20 | eqeltrid 2917 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
22 | 21 | ralrimiva 3182 | . 2 ⊢ (𝐴 ∈ dom vol → ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
23 | f1oi 6646 | . . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
24 | f1of 6609 | . . . . 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 24061 | . . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
27 | fss 6521 | . . . 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 24158 | . . 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 258 | 1 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 ⊆ wss 3935 𝒫 cpw 4537 I cid 5453 × cxp 5547 ◡ccnv 5548 dom cdm 5549 ran crn 5550 ↾ cres 5551 “ cima 5552 Fn wfn 6344 ⟶wf 6345 –1-1-onto→wf1o 6348 (class class class)co 7145 ℝcr 10525 ℝ*cxr 10663 (,)cioo 12728 volcvol 23993 MblFncmbf 24144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-er 8279 df-map 8398 df-pm 8399 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-sup 8895 df-inf 8896 df-oi 8963 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-n0 11887 df-z 11971 df-uz 12233 df-q 12338 df-rp 12380 df-xadd 12498 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-rlim 14836 df-sum 15033 df-xmet 20468 df-met 20469 df-ovol 23994 df-vol 23995 df-mbf 24149 |
This theorem is referenced by: (None) |
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