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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 5541 | . . . . 5 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = ((◡ I “ 𝑥) ∩ 𝐴) | |
2 | cnvi 5456 | . . . . . . . 8 ⊢ ◡ I = I | |
3 | 2 | imaeq1i 5382 | . . . . . . 7 ⊢ (◡ I “ 𝑥) = ( I “ 𝑥) |
4 | imai 5397 | . . . . . . 7 ⊢ ( I “ 𝑥) = 𝑥 | |
5 | 3, 4 | eqtri 2632 | . . . . . 6 ⊢ (◡ I “ 𝑥) = 𝑥 |
6 | 5 | ineq1i 3772 | . . . . 5 ⊢ ((◡ I “ 𝑥) ∩ 𝐴) = (𝑥 ∩ 𝐴) |
7 | 1, 6 | eqtri 2632 | . . . 4 ⊢ (◡( I ↾ 𝐴) “ 𝑥) = (𝑥 ∩ 𝐴) |
8 | ioof 12142 | . . . . . . 7 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
9 | ffn 5958 | . . . . . . 7 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → (,) Fn (ℝ* × ℝ*)) | |
10 | ovelrn 6708 | . . . . . . 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 23140 | . . . . . . . . 9 ⊢ (𝑦(,)𝑧) ∈ dom vol | |
14 | 12, 13 | syl6eqel 2696 | . . . . . . . 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 3018 | . . . . . 6 ⊢ (∃𝑦 ∈ ℝ* ∃𝑧 ∈ ℝ* 𝑥 = (𝑦(,)𝑧) → 𝑥 ∈ dom vol) |
17 | 11, 16 | sylbi 206 | . . . . 5 ⊢ (𝑥 ∈ ran (,) → 𝑥 ∈ dom vol) |
18 | id 22 | . . . . 5 ⊢ (𝐴 ∈ dom vol → 𝐴 ∈ dom vol) | |
19 | inmbl 23117 | . . . . 5 ⊢ ((𝑥 ∈ dom vol ∧ 𝐴 ∈ dom vol) → (𝑥 ∩ 𝐴) ∈ dom vol) | |
20 | 17, 18, 19 | syl2anr 494 | . . . 4 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (𝑥 ∩ 𝐴) ∈ dom vol) |
21 | 7, 20 | syl5eqel 2692 | . . 3 ⊢ ((𝐴 ∈ dom vol ∧ 𝑥 ∈ ran (,)) → (◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
22 | 21 | ralrimiva 2949 | . 2 ⊢ (𝐴 ∈ dom vol → ∀𝑥 ∈ ran (,)(◡( I ↾ 𝐴) “ 𝑥) ∈ dom vol) |
23 | f1oi 6086 | . . . . 5 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
24 | f1of 6050 | . . . . 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 23106 | . . . 4 ⊢ (𝐴 ∈ dom vol → 𝐴 ⊆ ℝ) | |
27 | fss 5969 | . . . 4 ⊢ ((( I ↾ 𝐴):𝐴⟶𝐴 ∧ 𝐴 ⊆ ℝ) → ( I ↾ 𝐴):𝐴⟶ℝ) | |
28 | 25, 26, 27 | sylancr 694 | . . 3 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴):𝐴⟶ℝ) |
29 | ismbf 23203 | . . 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 246 | 1 ⊢ (𝐴 ∈ dom vol → ( I ↾ 𝐴) ∈ MblFn) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 I cid 4948 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 (class class class)co 6549 ℝcr 9814 ℝ*cxr 9952 (,)cioo 12046 volcvol 23039 MblFncmbf 23189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-q 11665 df-rp 11709 df-xadd 11823 df-ioo 12050 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-xmet 19560 df-met 19561 df-ovol 23040 df-vol 23041 df-mbf 23194 |
This theorem is referenced by: (None) |
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