Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > elunirnmbfm | Structured version Visualization version GIF version |
Description: The property of being a measurable function. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
Ref | Expression |
---|---|
elunirnmbfm | ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mbfm 29640 | . . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}) | |
2 | 1 | mpt2fun 6660 | . . . 4 ⊢ Fun MblFnM |
3 | elunirn 6413 | . . . 4 ⊢ (Fun MblFnM → (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎))) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎)) |
5 | ovex 6577 | . . . . . 6 ⊢ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∈ V | |
6 | 5 | rabex 4740 | . . . . 5 ⊢ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} ∈ V |
7 | 1, 6 | dmmpt2 7129 | . . . 4 ⊢ dom MblFnM = (∪ ran sigAlgebra × ∪ ran sigAlgebra) |
8 | 7 | rexeqi 3120 | . . 3 ⊢ (∃𝑎 ∈ dom MblFnM𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎)) |
9 | fveq2 6103 | . . . . . 6 ⊢ (𝑎 = 〈𝑠, 𝑡〉 → (MblFnM‘𝑎) = (MblFnM‘〈𝑠, 𝑡〉)) | |
10 | df-ov 6552 | . . . . . 6 ⊢ (𝑠MblFnM𝑡) = (MblFnM‘〈𝑠, 𝑡〉) | |
11 | 9, 10 | syl6eqr 2662 | . . . . 5 ⊢ (𝑎 = 〈𝑠, 𝑡〉 → (MblFnM‘𝑎) = (𝑠MblFnM𝑡)) |
12 | 11 | eleq2d 2673 | . . . 4 ⊢ (𝑎 = 〈𝑠, 𝑡〉 → (𝐹 ∈ (MblFnM‘𝑎) ↔ 𝐹 ∈ (𝑠MblFnM𝑡))) |
13 | 12 | rexxp 5186 | . . 3 ⊢ (∃𝑎 ∈ (∪ ran sigAlgebra × ∪ ran sigAlgebra)𝐹 ∈ (MblFnM‘𝑎) ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡)) |
14 | 4, 8, 13 | 3bitri 285 | . 2 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡)) |
15 | simpl 472 | . . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑠 ∈ ∪ ran sigAlgebra) | |
16 | simpr 476 | . . . 4 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → 𝑡 ∈ ∪ ran sigAlgebra) | |
17 | 15, 16 | ismbfm 29641 | . . 3 ⊢ ((𝑠 ∈ ∪ ran sigAlgebra ∧ 𝑡 ∈ ∪ ran sigAlgebra) → (𝐹 ∈ (𝑠MblFnM𝑡) ↔ (𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠))) |
18 | 17 | 2rexbiia 3037 | . 2 ⊢ (∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra𝐹 ∈ (𝑠MblFnM𝑡) ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) |
19 | 14, 18 | bitri 263 | 1 ⊢ (𝐹 ∈ ∪ ran MblFnM ↔ ∃𝑠 ∈ ∪ ran sigAlgebra∃𝑡 ∈ ∪ ran sigAlgebra(𝐹 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∧ ∀𝑥 ∈ 𝑡 (◡𝐹 “ 𝑥) ∈ 𝑠)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 {crab 2900 〈cop 4131 ∪ cuni 4372 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 Fun wfun 5798 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 sigAlgebracsiga 29497 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-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 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-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-mbfm 29640 |
This theorem is referenced by: mbfmfun 29643 isanmbfm 29645 |
Copyright terms: Public domain | W3C validator |