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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 6182 | . . . . 5 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = ((◡ I “ 𝑥) ∩ 𝐴) | |
2 | cnvi 6094 | . . . . . . . 8 ⊢ ◡ I = I | |
3 | 2 | imaeq1i 6010 | . . . . . . 7 ⊢ (◡ I “ 𝑥) = ( I “ 𝑥) |
4 | imai 6026 | . . . . . . 7 ⊢ ( I “ 𝑥) = 𝑥 | |
5 | 3, 4 | eqtri 2764 | . . . . . 6 ⊢ (◡ I “ 𝑥) = 𝑥 |
6 | 5 | ineq1i 4168 | . . . . 5 ⊢ ((◡ I “ 𝑥) ∩ 𝐴) = (𝑥 ∩ 𝐴) |
7 | 1, 6 | eqtri 2764 | . . . 4 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = (𝑥 ∩ 𝐴) |
8 | ioof 13363 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
9 | ffn 6668 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
10 | ovelrn 7529 | . . . . . . 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 24927 | . . . . . . . . 9 ⊢ (𝑦(,)𝑧) ∈ dom vol | |
14 | 12, 13 | eqeltrdi 2846 | . . . . . . . 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 3196 | . . . . . 6 ⊢ (∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol) |
17 | 11, 16 | sylbi 216 | . . . . 5 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ dom vol) |
18 | id 22 | . . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ∈ dom vol) | |
19 | inmbl 24904 | . . . . 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 2842 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
22 | 21 | ralrimiva 3143 | . 2 ⊢ (𝐴 ∈ dom vol → ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
23 | f1oi 6822 | . . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
24 | f1of 6784 | . . . . 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 24893 | . . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
27 | fss 6685 | . . . 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 24990 | . . 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 256 | 1 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∃wrex 3073 ∩ cin 3909 ⊆ wss 3910 𝒫 cpw 4560 I cid 5530 × cxp 5631 ◡ccnv 5632 dom cdm 5633 ran crn 5634 ↾ cres 5635 “ cima 5636 Fn wfn 6491 ⟶wf 6492 –1-1-onto→wf1o 6495 (class class class)co 7356 ℝcr 11049 ℝ*cxr 11187 (,)cioo 13263 volcvol 24825 MblFncmbf 24976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-sup 9377 df-inf 9378 df-oi 9445 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-z 12499 df-uz 12763 df-q 12873 df-rp 12915 df-xadd 13033 df-ioo 13267 df-ico 13269 df-icc 13270 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-hash 14230 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-clim 15369 df-rlim 15370 df-sum 15570 df-xmet 20787 df-met 20788 df-ovol 24826 df-vol 24827 df-mbf 24981 |
This theorem is referenced by: (None) |
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