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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 6049 | . . 3 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝐴)) | |
2 | 1 | eleq1d 2812 | . 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 33779 | . . . 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 3607 | 1 ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∪ cuni 4902 ◡ccnv 5668 ran crn 5670 “ cima 5672 (class class class)co 7405 ↑m cmap 8822 sigAlgebracsiga 33636 MblFnMcmbfm 33777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-mbfm 33778 |
This theorem is referenced by: imambfm 33791 mbfmco 33793 mbfmco2 33794 sxbrsiga 33819 sibfinima 33868 sibfof 33869 orvcoel 33990 orvccel 33991 |
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