mbff - Metamath Proof Explorer

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Theorem mbff 25953

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 25952	. . 3 ⊢ (𝐹 ∈ MblFn ↔ (𝐹 ∈ (ℂ ↑_pm ℝ) ∧ ∀𝑥 ∈ ran (,)((^◡(ℜ ∘ 𝐹) “ 𝑥) ∈ dom vol ∧ (^◡(ℑ ∘ 𝐹) “ 𝑥) ∈ dom vol)))
2	1	simplbi 502	. 2 ⊢ (𝐹 ∈ MblFn → 𝐹 ∈ (ℂ ↑_pm ℝ))
3		cnex 11286	. . . 4 ⊢ ℂ ∈ V
4		reex 11296	. . . 4 ⊢ ℝ ∈ V
5	3, 4	elpm2 8907	. . 3 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
6	5	simplbi 502	. 2 ⊢ (𝐹 ∈ (ℂ ↑_pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
7	2, 6	syl 18	1 ⊢ (𝐹 ∈ MblFn → 𝐹:dom 𝐹⟶ℂ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2144 ∀wral 3076 ⊆ wss 3896 ^◡ccnv 5649 dom cdm 5650 ran crn 5651 “ cima 5653 ∘ ccom 5654 ⟶wf 6537 (class class class)co 7422 ↑_pm cpm 8853 ℂcc 11203 ℝcr 11204 (,)cioo 13481 ℜcre 15271 ℑcim 15272 volcvol 25791 MblFncmbf 25942

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7425 df-oprab 7426 df-mpo 7427 df-pm 8855 df-mbf 25947

This theorem is used by: mbfdm 25954 mbfmptcl 25964 mbfres 25972 mbfimaopnlem 25983 mbfadd 25989 mbfsub 25990 mbfmul 26054 iblcnlem 26116 bddmulibl 26166 bddibl 26167 bddiblnc 26169 mbfresmf 47856

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