Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnmbfm | Structured version Visualization version GIF version |
Description: A continuous function is measurable with respect to the Borel Algebra of its domain and range. (Contributed by Thierry Arnoux, 3-Jun-2017.) |
Ref | Expression |
---|---|
cnmbfm.1 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
cnmbfm.2 | ⊢ (𝜑 → 𝑆 = (sigaGen‘𝐽)) |
cnmbfm.3 | ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) |
Ref | Expression |
---|---|
cnmbfm | ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnmbfm.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
2 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | eqid 2610 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 20860 | . . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾) |
6 | cnmbfm.2 | . . . . . 6 ⊢ (𝜑 → 𝑆 = (sigaGen‘𝐽)) | |
7 | 6 | unieqd 4382 | . . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ (sigaGen‘𝐽)) |
8 | cntop1 20854 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
9 | unisg 29533 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
10 | 1, 8, 9 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
11 | 7, 10 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
12 | cnmbfm.3 | . . . . . 6 ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) | |
13 | 12 | unieqd 4382 | . . . . 5 ⊢ (𝜑 → ∪ 𝑇 = ∪ (sigaGen‘𝐾)) |
14 | cntop2 20855 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
15 | unisg 29533 | . . . . . 6 ⊢ (𝐾 ∈ Top → ∪ (sigaGen‘𝐾) = ∪ 𝐾) | |
16 | 1, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘𝐾) = ∪ 𝐾) |
17 | 13, 16 | eqtrd 2644 | . . . 4 ⊢ (𝜑 → ∪ 𝑇 = ∪ 𝐾) |
18 | 11, 17 | feq23d 5953 | . . 3 ⊢ (𝜑 → (𝐹:∪ 𝑆⟶∪ 𝑇 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
19 | 5, 18 | mpbird 246 | . 2 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) |
20 | sssigagen 29535 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (sigaGen‘𝐽)) | |
21 | 1, 8, 20 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ (sigaGen‘𝐽)) |
22 | 21, 6 | sseqtr4d 3605 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
23 | 22 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → 𝐽 ⊆ 𝑆) |
24 | cnima 20879 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝐽) | |
25 | 1, 24 | sylan 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝐽) |
26 | 23, 25 | sseldd 3569 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝑆) |
27 | 26 | ralrimiva 2949 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆) |
28 | elex 3185 | . . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ∈ V) | |
29 | 1, 14, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐾 ∈ V) |
30 | sigagensiga 29531 | . . . . . 6 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) | |
31 | 1, 8, 30 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) |
32 | 6, 31 | eqeltrd 2688 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝐽)) |
33 | elrnsiga 29516 | . . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝐽) → 𝑆 ∈ ∪ ran sigAlgebra) | |
34 | 32, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
35 | 29, 34, 12 | imambfm 29651 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆))) |
36 | 19, 27, 35 | mpbir2and 959 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 ◡ccnv 5037 ran crn 5039 “ cima 5041 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Topctop 20517 Cn ccn 20838 sigAlgebracsiga 29497 sigaGencsigagen 29528 MblFnMcmbfm 29639 |
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-ac2 9168 |
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-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-iin 4458 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-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-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-oi 8298 df-card 8648 df-acn 8651 df-ac 8822 df-cda 8873 df-top 20521 df-topon 20523 df-cn 20841 df-siga 29498 df-sigagen 29529 df-mbfm 29640 |
This theorem is referenced by: sxbrsiga 29679 rrvadd 29841 rrvmulc 29842 |
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