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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 23918 | . . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∀𝑥 ∈ ran (,)((◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol))) | |
2 | 1 | simplbi 490 | . 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10408 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10418 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8230 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simplbi 490 | . 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 387 ∈ wcel 2048 ∀wral 3082 ⊆ wss 3825 ◡ccnv 5399 dom cdm 5400 ran crn 5401 “ cima 5403 ∘ ccom 5404 ⟶wf 6178 (class class class)co 6970 ↑pm cpm 8199 ℂcc 10325 ℝcr 10326 (,)cioo 12547 ℜcre 14307 ℑcim 14308 volcvol 23757 MblFncmbf 23908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-id 5305 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-fv 6190 df-ov 6973 df-oprab 6974 df-mpo 6975 df-pm 8201 df-mbf 23913 |
This theorem is referenced by: mbfdm 23920 mbfmptcl 23930 mbfres 23938 mbfimaopnlem 23949 mbfadd 23955 mbfsub 23956 mbfmul 24020 iblcnlem 24082 bddmulibl 24132 bddibl 24133 bddiblnc 34351 mbfresmf 42393 |
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