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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 6014 | . . 3 ⊢ (𝑥 = 𝐴 → (◡𝐹 “ 𝑥) = (◡𝐹 “ 𝐴)) | |
2 | 1 | eleq1d 2823 | . 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 32890 | . . . 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 497 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑇 (◡𝐹 “ 𝑥) ∈ 𝑆) |
9 | mbfmcnvima.4 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑇) | |
10 | 2, 8, 9 | rspcdva 3585 | 1 ⊢ (𝜑 → (◡𝐹 “ 𝐴) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∪ cuni 4870 ◡ccnv 5637 ran crn 5639 “ cima 5641 (class class class)co 7362 ↑m cmap 8772 sigAlgebracsiga 32747 MblFnMcmbfm 32888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-mbfm 32889 |
This theorem is referenced by: imambfm 32902 mbfmco 32904 mbfmco2 32905 sxbrsiga 32930 sibfinima 32979 sibfof 32980 orvcoel 33101 orvccel 33102 |
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