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 2823 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | eqid 2823 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 21856 | . . . 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 4854 | . . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ (sigaGen‘𝐽)) |
8 | cntop1 21850 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
9 | unisg 31404 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
10 | 1, 8, 9 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
11 | 7, 10 | eqtrd 2858 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
12 | cnmbfm.3 | . . . . . 6 ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) | |
13 | 12 | unieqd 4854 | . . . . 5 ⊢ (𝜑 → ∪ 𝑇 = ∪ (sigaGen‘𝐾)) |
14 | cntop2 21851 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
15 | unisg 31404 | . . . . . 6 ⊢ (𝐾 ∈ Top → ∪ (sigaGen‘𝐾) = ∪ 𝐾) | |
16 | 1, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘𝐾) = ∪ 𝐾) |
17 | 13, 16 | eqtrd 2858 | . . . 4 ⊢ (𝜑 → ∪ 𝑇 = ∪ 𝐾) |
18 | 11, 17 | feq23d 6511 | . . 3 ⊢ (𝜑 → (𝐹:∪ 𝑆⟶∪ 𝑇 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
19 | 5, 18 | mpbird 259 | . 2 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) |
20 | sssigagen 31406 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (sigaGen‘𝐽)) | |
21 | 1, 8, 20 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ (sigaGen‘𝐽)) |
22 | 21, 6 | sseqtrrd 4010 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
23 | 22 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → 𝐽 ⊆ 𝑆) |
24 | cnima 21875 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝐽) | |
25 | 1, 24 | sylan 582 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝐽) |
26 | 23, 25 | sseldd 3970 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝑆) |
27 | 26 | ralrimiva 3184 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆) |
28 | elex 3514 | . . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ∈ V) | |
29 | 1, 14, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐾 ∈ V) |
30 | sigagensiga 31402 | . . . . . 6 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) | |
31 | 1, 8, 30 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) |
32 | 6, 31 | eqeltrd 2915 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝐽)) |
33 | elrnsiga 31387 | . . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝐽) → 𝑆 ∈ ∪ ran sigAlgebra) | |
34 | 32, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
35 | 29, 34, 12 | imambfm 31522 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆))) |
36 | 19, 27, 35 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ⊆ wss 3938 ∪ cuni 4840 ◡ccnv 5556 ran crn 5558 “ cima 5560 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Topctop 21503 Cn ccn 21834 sigAlgebracsiga 31369 sigaGencsigagen 31399 MblFnMcmbfm 31510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-ac2 9887 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-oi 8976 df-dju 9332 df-card 9370 df-acn 9373 df-ac 9544 df-top 21504 df-topon 21521 df-cn 21837 df-siga 31370 df-sigagen 31400 df-mbfm 31511 |
This theorem is referenced by: sxbrsiga 31550 rrvadd 31712 rrvmulc 31713 |
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