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 31885 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
5 | 3, 4 | mpbid 235 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
6 | unieq 4816 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
7 | 6 | oveq2d 7207 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑m ∪ 𝑠) = (∪ 𝑡 ↑m ∪ 𝑆)) |
8 | 7 | eleq2d 2816 | . . . . 5 ⊢ (𝑠 = 𝑆 → (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ↔ 𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆))) |
9 | eleq2 2819 | . . . . . 6 ⊢ (𝑠 = 𝑆 → ((◡𝐹 “ 𝑥) ∈ 𝑠 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆)) | |
10 | 9 | ralbidv 3108 | . . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
11 | 8, 10 | anbi12d 634 | . . . 4 ⊢ (𝑠 = 𝑆 → ((𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠) ↔ (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
12 | unieq 4816 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇) | |
13 | 12 | oveq1d 7206 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑m ∪ 𝑆) = (∪ 𝑇 ↑m ∪ 𝑆)) |
14 | 13 | eleq2d 2816 | . . . . 5 ⊢ (𝑡 = 𝑇 → (𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑆) ↔ 𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆))) |
15 | raleq 3309 | . . . . 5 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) | |
16 | 14, 15 | anbi12d 634 | . . . 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 3539 | . . 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 1373 | . 2 ⊢ (𝜑 → ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) |
19 | elunirnmbfm 31886 | . 2 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑m ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) | |
20 | 18, 19 | sylibr 237 | 1 ⊢ (𝜑 → 𝐹 ∈ ∪ ran MblFnM) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ∪ cuni 4805 ◡ccnv 5535 ran crn 5537 “ cima 5539 (class class class)co 7191 ↑m cmap 8486 sigAlgebracsiga 31742 MblFnMcmbfm 31883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-mbfm 31884 |
This theorem is referenced by: mbfmbfm 31891 orvcval4 32093 |
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