Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ismbfm | Structured version Visualization version GIF version |
Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23397. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
Ref | Expression |
---|---|
ismbfm.1 | ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
ismbfm.2 | ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) |
Ref | Expression |
---|---|
ismbfm | ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbfm.1 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
2 | ismbfm.2 | . . . 4 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) | |
3 | unieq 4444 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
4 | 3 | oveq2d 6666 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑𝑚 ∪ 𝑠) = (∪ 𝑡 ↑𝑚 ∪ 𝑆)) |
5 | eleq2 2690 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ 𝑥) ∈ 𝑠 ↔ (◡𝑓 “ 𝑥) ∈ 𝑆)) | |
6 | 5 | ralbidv 2986 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆)) |
7 | 4, 6 | rabeqbidv 3195 | . . . . 5 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} = {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
8 | unieq 4444 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇) | |
9 | 8 | oveq1d 6665 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑𝑚 ∪ 𝑆) = (∪ 𝑇 ↑𝑚 ∪ 𝑆)) |
10 | raleq 3138 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆)) | |
11 | 9, 10 | rabeqbidv 3195 | . . . . 5 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆} = {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
12 | df-mbfm 30313 | . . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}) | |
13 | ovex 6678 | . . . . . 6 ⊢ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∈ V | |
14 | 13 | rabex 4813 | . . . . 5 ⊢ {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ∈ V |
15 | 7, 11, 12, 14 | ovmpt2 6796 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
16 | 1, 2, 15 | syl2anc 693 | . . 3 ⊢ (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
17 | 16 | eleq2d 2687 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})) |
18 | cnveq 5296 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
19 | 18 | imaeq1d 5465 | . . . . 5 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑥) = (◡𝐹 “ 𝑥)) |
20 | 19 | eleq1d 2686 | . . . 4 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆)) |
21 | 20 | ralbidv 2986 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
22 | 21 | elrab 3363 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ↔ (𝐹 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
23 | 17, 22 | syl6bb 276 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ∪ cuni 4436 ◡ccnv 5113 ran crn 5115 “ cima 5117 (class class class)co 6650 ↑𝑚 cmap 7857 sigAlgebracsiga 30170 MblFnMcmbfm 30312 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-mbfm 30313 |
This theorem is referenced by: elunirnmbfm 30315 mbfmf 30317 isanmbfm 30318 mbfmcnvima 30319 mbfmcst 30321 1stmbfm 30322 2ndmbfm 30323 imambfm 30324 mbfmco 30326 elmbfmvol2 30329 mbfmcnt 30330 sibfof 30402 isrrvv 30505 |
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