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Theorem mbfmcst 34261
Description: A constant function is measurable. Cf. mbfconst 25668. (Contributed by Thierry Arnoux, 26-Jan-2017.)
Hypotheses
Ref Expression
mbfmcst.1 (𝜑𝑆 ran sigAlgebra)
mbfmcst.2 (𝜑𝑇 ran sigAlgebra)
mbfmcst.3 (𝜑𝐹 = (𝑥 𝑆𝐴))
mbfmcst.4 (𝜑𝐴 𝑇)
Assertion
Ref Expression
mbfmcst (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑆   𝑥,𝑇   𝜑,𝑥
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem mbfmcst
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 mbfmcst.3 . . . 4 (𝜑𝐹 = (𝑥 𝑆𝐴))
2 mbfmcst.4 . . . . 5 (𝜑𝐴 𝑇)
32adantr 480 . . . 4 ((𝜑𝑥 𝑆) → 𝐴 𝑇)
41, 3fmpt3d 7136 . . 3 (𝜑𝐹: 𝑆 𝑇)
5 mbfmcst.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
6 unielsiga 34129 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
75, 6syl 17 . . . 4 (𝜑 𝑇𝑇)
8 mbfmcst.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
9 unielsiga 34129 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
108, 9syl 17 . . . 4 (𝜑 𝑆𝑆)
117, 10elmapd 8880 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ↔ 𝐹: 𝑆 𝑇))
124, 11mpbird 257 . 2 (𝜑𝐹 ∈ ( 𝑇m 𝑆))
13 fconstmpt 5747 . . . . . . . . . . 11 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
1413cnveqi 5885 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
15 cnvxp 6177 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = ({𝐴} × 𝑆)
1614, 15eqtr3i 2767 . . . . . . . . 9 (𝑥 𝑆𝐴) = ({𝐴} × 𝑆)
1716imaeq1i 6075 . . . . . . . 8 ((𝑥 𝑆𝐴) “ 𝑦) = (({𝐴} × 𝑆) “ 𝑦)
18 df-ima 5698 . . . . . . . 8 (({𝐴} × 𝑆) “ 𝑦) = ran (({𝐴} × 𝑆) ↾ 𝑦)
19 df-rn 5696 . . . . . . . 8 ran (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
2017, 18, 193eqtri 2769 . . . . . . 7 ((𝑥 𝑆𝐴) “ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
21 df-res 5697 . . . . . . . . . 10 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} × 𝑆) ∩ (𝑦 × V))
22 inxp 5842 . . . . . . . . . 10 (({𝐴} × 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V))
23 inv1 4398 . . . . . . . . . . 11 ( 𝑆 ∩ V) = 𝑆
2423xpeq2i 5712 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × 𝑆)
2521, 22, 243eqtri 2769 . . . . . . . . 9 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2625cnveqi 5885 . . . . . . . 8 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2726dmeqi 5915 . . . . . . 7 dom (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} ∩ 𝑦) × 𝑆)
28 cnvxp 6177 . . . . . . . 8 (({𝐴} ∩ 𝑦) × 𝑆) = ( 𝑆 × ({𝐴} ∩ 𝑦))
2928dmeqi 5915 . . . . . . 7 dom (({𝐴} ∩ 𝑦) × 𝑆) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
3020, 27, 293eqtri 2769 . . . . . 6 ((𝑥 𝑆𝐴) “ 𝑦) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
31 xpeq2 5706 . . . . . . . . . . 11 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ( 𝑆 × ∅))
32 xp0 6178 . . . . . . . . . . 11 ( 𝑆 × ∅) = ∅
3331, 32eqtrdi 2793 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3433dmeqd 5916 . . . . . . . . 9 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35 dm0 5931 . . . . . . . . 9 dom ∅ = ∅
3634, 35eqtrdi 2793 . . . . . . . 8 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3736adantl 481 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38 0elsiga 34115 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
398, 38syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑆)
4039adantr 480 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
4137, 40eqeltrd 2841 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4230, 41eqeltrid 2845 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
43 dmxp 5939 . . . . . . . 8 (({𝐴} ∩ 𝑦) ≠ ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4443adantl 481 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4510adantr 480 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → 𝑆𝑆)
4644, 45eqeltrd 2841 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4730, 46eqeltrid 2845 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4842, 47pm2.61dane 3029 . . . 4 (𝜑 → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4948ralrimivw 3150 . . 3 (𝜑 → ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
501cnveqd 5886 . . . . . 6 (𝜑𝐹 = (𝑥 𝑆𝐴))
5150imaeq1d 6077 . . . . 5 (𝜑 → (𝐹𝑦) = ((𝑥 𝑆𝐴) “ 𝑦))
5251eleq1d 2826 . . . 4 (𝜑 → ((𝐹𝑦) ∈ 𝑆 ↔ ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5352ralbidv 3178 . . 3 (𝜑 → (∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5449, 53mpbird 257 . 2 (𝜑 → ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)
558, 5ismbfm 34252 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)))
5612, 54, 55mpbir2and 713 1 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cin 3950  c0 4333  {csn 4626   cuni 4907  cmpt 5225   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688  wf 6557  (class class class)co 7431  m cmap 8866  sigAlgebracsiga 34109  MblFnMcmbfm 34250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-siga 34110  df-mbfm 34251
This theorem is referenced by:  sibf0  34336
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