Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 2818 | . . . . 5 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
3 | eqid 2818 | . . . . 5 ⊢ ∪ 𝐾 = ∪ 𝐾 | |
4 | 2, 3 | cnf 21782 | . . . 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 4840 | . . . . 5 ⊢ (𝜑 → ∪ 𝑆 = ∪ (sigaGen‘𝐽)) |
8 | cntop1 21776 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top) | |
9 | unisg 31301 | . . . . . 6 ⊢ (𝐽 ∈ Top → ∪ (sigaGen‘𝐽) = ∪ 𝐽) | |
10 | 1, 8, 9 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽) |
11 | 7, 10 | eqtrd 2853 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝐽) |
12 | cnmbfm.3 | . . . . . 6 ⊢ (𝜑 → 𝑇 = (sigaGen‘𝐾)) | |
13 | 12 | unieqd 4840 | . . . . 5 ⊢ (𝜑 → ∪ 𝑇 = ∪ (sigaGen‘𝐾)) |
14 | cntop2 21777 | . . . . . 6 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top) | |
15 | unisg 31301 | . . . . . 6 ⊢ (𝐾 ∈ Top → ∪ (sigaGen‘𝐾) = ∪ 𝐾) | |
16 | 1, 14, 15 | 3syl 18 | . . . . 5 ⊢ (𝜑 → ∪ (sigaGen‘𝐾) = ∪ 𝐾) |
17 | 13, 16 | eqtrd 2853 | . . . 4 ⊢ (𝜑 → ∪ 𝑇 = ∪ 𝐾) |
18 | 11, 17 | feq23d 6502 | . . 3 ⊢ (𝜑 → (𝐹:∪ 𝑆⟶∪ 𝑇 ↔ 𝐹:∪ 𝐽⟶∪ 𝐾)) |
19 | 5, 18 | mpbird 258 | . 2 ⊢ (𝜑 → 𝐹:∪ 𝑆⟶∪ 𝑇) |
20 | sssigagen 31303 | . . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ⊆ (sigaGen‘𝐽)) | |
21 | 1, 8, 20 | 3syl 18 | . . . . . 6 ⊢ (𝜑 → 𝐽 ⊆ (sigaGen‘𝐽)) |
22 | 21, 6 | sseqtrrd 4005 | . . . . 5 ⊢ (𝜑 → 𝐽 ⊆ 𝑆) |
23 | 22 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → 𝐽 ⊆ 𝑆) |
24 | cnima 21801 | . . . . 5 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝐽) | |
25 | 1, 24 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝐽) |
26 | 23, 25 | sseldd 3965 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐾) → (◡𝐹 “ 𝑎) ∈ 𝑆) |
27 | 26 | ralrimiva 3179 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆) |
28 | elex 3510 | . . . 4 ⊢ (𝐾 ∈ Top → 𝐾 ∈ V) | |
29 | 1, 14, 28 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐾 ∈ V) |
30 | sigagensiga 31299 | . . . . . 6 ⊢ (𝐽 ∈ Top → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) | |
31 | 1, 8, 30 | 3syl 18 | . . . . 5 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ (sigAlgebra‘∪ 𝐽)) |
32 | 6, 31 | eqeltrd 2910 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ (sigAlgebra‘∪ 𝐽)) |
33 | elrnsiga 31284 | . . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝐽) → 𝑆 ∈ ∪ ran sigAlgebra) | |
34 | 32, 33 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
35 | 29, 34, 12 | imambfm 31419 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹:∪ 𝑆⟶∪ 𝑇 ∧ ∀𝑎 ∈ 𝐾 (◡𝐹 “ 𝑎) ∈ 𝑆))) |
36 | 19, 27, 35 | mpbir2and 709 | 1 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ∪ cuni 4830 ◡ccnv 5547 ran crn 5549 “ cima 5551 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 Topctop 21429 Cn ccn 21760 sigAlgebracsiga 31266 sigaGencsigagen 31296 MblFnMcmbfm 31407 |
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 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-ac2 9873 |
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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-ac 9530 df-top 21430 df-topon 21447 df-cn 21763 df-siga 31267 df-sigagen 31297 df-mbfm 31408 |
This theorem is referenced by: sxbrsiga 31447 rrvadd 31609 rrvmulc 31610 |
Copyright terms: Public domain | W3C validator |