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Mirrors > Home > MPE Home > Th. List > mbff | Structured version Visualization version GIF version |
Description: A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.) |
Ref | Expression |
---|---|
mbff | ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismbf1 24227 | . . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
2 | 1 | simplbi 500 | . 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10620 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10630 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8440 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simplbi 500 | . 2 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℂ) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 ⊆ wss 3938 ◡ccnv 5556 dom cdm 5557 ran crn 5558 “ cima 5560 ∘ ccom 5561 ⟶wf 6353 (class class class)co 7158 ↑pm cpm 8409 ℂcc 10537 ℝcr 10538 (,)cioo 12741 ℜcre 14458 ℑcim 14459 volcvol 24066 MblFncmbf 24217 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-pm 8411 df-mbf 24222 |
This theorem is referenced by: mbfdm 24229 mbfmptcl 24239 mbfres 24247 mbfimaopnlem 24258 mbfadd 24264 mbfsub 24265 mbfmul 24329 iblcnlem 24391 bddmulibl 24441 bddibl 24442 bddiblnc 34964 mbfresmf 43023 |
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