mbff - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mbff		Structured version Visualization version GIF version

Theorem mbff 25592

Description: A measurable function is a function into the complex numbers. (Contributed by Mario Carneiro, 17-Jun-2014.)

**Assertion**
Ref	Expression
mbff	⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Proof of Theorem mbff Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismbf1 25591	. . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
2	1	simplbi 496	. 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11119	. . . 4 ⊢ ℂ ∈ V
4		reex 11129	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8822	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 496	. 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 395 ∈ wcel 2114 ∀wral 3051 ⊆ wss 3889 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 “ cima 5634 ∘ ccom 5635 ⟶wf 6494 (class class class)co 7367 ↑_pm cpm 8774 ℂcc 11036 ℝcr 11037 (,)cioo 13298 ℜcre 15059 ℑcim 15060 volcvol 25430 MblFncmbf 25581

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-pm 8776 df-mbf 25586

This theorem is referenced by: mbfdm 25593 mbfmptcl 25603 mbfres 25611 mbfimaopnlem 25622 mbfadd 25628 mbfsub 25629 mbfmul 25693 iblcnlem 25756 bddmulibl 25806 bddibl 25807 bddiblnc 25809 mbfresmf 47167

W3C validator