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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mbfmcnvima | Structured version Visualization version GIF version |
Description: The preimage by a measurable function is a measurable set. (Contributed by Thierry Arnoux, 23-Jan-2017.) |
Ref | Expression |
---|---|
mbfmf.1 | ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) |
mbfmf.2 | ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) |
mbfmf.3 | ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) |
mbfmcnvima.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑇) |
Ref | Expression |
---|---|
mbfmcnvima | ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq2 6055 | . . 3 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝐴)) | |
2 | 1 | eleq1d 2817 | . 2 ⊢ (𝑥 = 𝐴 → ((◡𝐹 “ 𝑥) ∈ 𝑆 ↔ (◡𝐹 “ 𝐴) ∈ 𝑆)) |
3 | mbfmf.3 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑆MblFnM𝑇)) | |
4 | mbfmf.1 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra) | |
5 | mbfmf.2 | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ ∪ ran sigAlgebra) | |
6 | 4, 5 | ismbfm 33712 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆))) |
7 | 3, 6 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (∪ 𝑇 ↑m ∪ 𝑆) ∧ ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆)) |
8 | 7 | simprd 495 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆) |
9 | mbfmcnvima.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
10 | 2, 8, 9 | rspcdva 3613 | 1 ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 ∀wral 3060 ∪ cuni 4908 ◡ccnv 5675 ran crn 5677 “ cima 5679 (class class class)co 7412 ↑m cmap 8826 sigAlgebracsiga 33569 MblFnMcmbfm 33710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-mbfm 33711 |
This theorem is referenced by: imambfm 33724 mbfmco 33726 mbfmco2 33727 sxbrsiga 33752 sibfinima 33801 sibfof 33802 orvcoel 33923 orvccel 33924 |
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