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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfdmss | Structured version Visualization version GIF version |
Description: The domain of a function measurable w.r.t. to a sigma-algebra, is a subset of the set underlying the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfdmss.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfdmss.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfdmss.d | ⊢ 𝐷 = dom 𝐹 |
Ref | Expression |
---|---|
smfdmss | ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfdmss.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
2 | smfdmss.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
3 | smfdmss.d | . . . 4 ⊢ 𝐷 = dom 𝐹 | |
4 | 2, 3 | issmf 46684 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
5 | 1, 4 | mpbid 232 | . 2 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
6 | 5 | simp1d 1141 | 1 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 < clt 11293 ↾t crest 17467 SAlgcsalg 46264 SMblFncsmblfn 46651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388 df-ico 13390 df-smblfn 46652 |
This theorem is referenced by: sssmf 46694 smfsssmf 46699 issmfle 46701 smfpimltxr 46703 issmfgt 46712 smfadd 46721 issmfge 46726 smflim 46733 smfpimgtxr 46736 smfpimioo 46743 smfresal 46744 smfrec 46745 smfres 46746 smfmul 46751 smfmulc1 46752 smfco 46758 smfsuplem3 46769 smfpimne 46795 smfpimne2 46796 |
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