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 23203. (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 4380 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
4 | 3 | oveq2d 6565 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑡 ↑𝑚 ∪ 𝑠) = (∪ 𝑡 ↑𝑚 ∪ 𝑆)) |
5 | eleq2 2677 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ((◡𝑓 “ 𝑥) ∈ 𝑠 ↔ (◡𝑓 “ 𝑥) ∈ 𝑆)) | |
6 | 5 | ralbidv 2969 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠 ↔ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆)) |
7 | 4, 6 | rabeqbidv 3168 | . . . . 5 ⊢ (𝑠 = 𝑆 → {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠} = {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
8 | unieq 4380 | . . . . . . 7 ⊢ (𝑡 = 𝑇 → ∪ 𝑡 = ∪ 𝑇) | |
9 | 8 | oveq1d 6564 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (∪ 𝑡 ↑𝑚 ∪ 𝑆) = (∪ 𝑇 ↑𝑚 ∪ 𝑆)) |
10 | raleq 3115 | . . . . . 6 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆)) | |
11 | 9, 10 | rabeqbidv 3168 | . . . . 5 ⊢ (𝑡 = 𝑇 → {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑆} = {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
12 | df-mbfm 29640 | . . . . 5 ⊢ MblFnM = (𝑠 ∈ ∪ ran sigAlgebra, 𝑡 ∈ ∪ ran sigAlgebra ↦ {𝑓 ∈ (∪ 𝑡 ↑𝑚 ∪ 𝑠) ∣ ∀𝑥 ∈ 𝑡 (◡𝑓 “ 𝑥) ∈ 𝑠}) | |
13 | ovex 6577 | . . . . . 6 ⊢ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∈ V | |
14 | 13 | rabex 4740 | . . . . 5 ⊢ {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ∈ V |
15 | 7, 11, 12, 14 | ovmpt2 6694 | . . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
16 | 1, 2, 15 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆}) |
17 | 16 | eleq2d 2673 | . 2 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆})) |
18 | cnveq 5218 | . . . . . 6 ⊢ (𝑓 = 𝐹 → ◡𝑓 = ◡𝐹) | |
19 | 18 | imaeq1d 5384 | . . . . 5 ⊢ (𝑓 = 𝐹 → (◡𝑓 “ 𝑥) = (◡𝐹 “ 𝑥)) |
20 | 19 | eleq1d 2672 | . . . 4 ⊢ (𝑓 = 𝐹 → ((◡𝑓 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝑥) ∈ 𝑆)) |
21 | 20 | ralbidv 2969 | . . 3 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
22 | 21 | elrab 3331 | . 2 ⊢ (𝐹 ∈ {𝑓 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∣ ∀𝑥 ∈ 𝑇 (◡𝑓 “ 𝑥) ∈ 𝑆} ↔ (𝐹 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
23 | 17, 22 | syl6bb 275 | 1 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑𝑚 ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∪ cuni 4372 ◡ccnv 5037 ran crn 5039 “ cima 5041 (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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 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-br 4584 df-opab 4644 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-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-mbfm 29640 |
This theorem is referenced by: elunirnmbfm 29642 mbfmf 29644 isanmbfm 29645 mbfmcnvima 29646 mbfmcst 29648 1stmbfm 29649 2ndmbfm 29650 imambfm 29651 mbfmco 29653 elmbfmvol2 29656 mbfmcnt 29657 sibfof 29729 isrrvv 29832 |
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