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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 23608 | . . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
2 | 1 | simplbi 485 | . 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10219 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10229 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8041 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simplbi 485 | . 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 382 ∈ wcel 2145 ∀wral 3061 ⊆ wss 3723 ◡ccnv 5248 dom cdm 5249 ran crn 5250 “ cima 5252 ∘ ccom 5253 ⟶wf 6025 (class class class)co 6792 ↑pm cpm 8010 ℂcc 10136 ℝcr 10137 (,)cioo 12376 ℜcre 14041 ℑcim 14042 volcvol 23447 MblFncmbf 23598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-fv 6037 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-pm 8012 df-mbf 23603 |
This theorem is referenced by: mbfdm 23610 mbfmptcl 23620 mbfres 23627 mbfimaopnlem 23638 mbfadd 23644 mbfsub 23645 mbfmul 23709 iblcnlem 23771 bddmulibl 23821 bddibl 23822 bddiblnc 33808 mbfresmf 41465 |
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