Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > isanmbfm | Structured version Visualization version GIF version |
Description: The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.) |
Ref | Expression |
---|---|
mbfmf.1 | ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
mbfmf.2 | ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) |
mbfmf.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) |
Ref | Expression |
---|---|
isanmbfm | ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mbfmf.1 | . . 3 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
2 | mbfmf.2 | . . 3 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) | |
3 | mbfmf.3 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | |
4 | 1, 2 | ismbfm 31409 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
5 | 3, 4 | mpbid 233 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
6 | unieq 4838 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
7 | 6 | oveq2d 7161 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆)) |
8 | 7 | eleq2d 2895 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ↔ 𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆))) |
9 | eleq2 2898 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((◡𝐹 “ 𝑥) ∈ 𝑠 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆)) | |
10 | 9 | ralbidv 3194 | . . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
11 | 8, 10 | anbi12d 630 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) ↔ (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
12 | unieq 4838 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇) | |
13 | 12 | oveq1d 7160 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆)) |
14 | 13 | eleq2d 2895 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ↔ 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆))) |
15 | raleq 3403 | . . . . 5 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) | |
16 | 14, 15 | anbi12d 630 | . . . 4 ⊢ (𝑡 = 𝑇 → ((𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
17 | 11, 16 | rspc2ev 3632 | . . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra ∧ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) |
18 | 1, 2, 5, 17 | syl3anc 1363 | . 2 ⊢ (𝜑 → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) |
19 | elunirnmbfm 31410 | . 2 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) | |
20 | 18, 19 | sylibr 235 | 1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∪ cuni 4830 ◡ccnv 5547 ran crn 5549 “ cima 5551 (class class class)co 7145 ↑m cmap 8395 sigAlgebracsiga 31266 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-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 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-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 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-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-mbfm 31408 |
This theorem is referenced by: mbfmbfm 31415 orvcval4 31617 |
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