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Theorem mbfmcst 33327
Description: A constant function is measurable. Cf. mbfconst 25157. (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 481 . . . 4 ((𝜑𝑥 𝑆) → 𝐴 𝑇)
41, 3fmpt3d 7117 . . 3 (𝜑𝐹: 𝑆 𝑇)
5 mbfmcst.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
6 unielsiga 33195 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
75, 6syl 17 . . . 4 (𝜑 𝑇𝑇)
8 mbfmcst.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
9 unielsiga 33195 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
108, 9syl 17 . . . 4 (𝜑 𝑆𝑆)
117, 10elmapd 8836 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ↔ 𝐹: 𝑆 𝑇))
124, 11mpbird 256 . 2 (𝜑𝐹 ∈ ( 𝑇m 𝑆))
13 fconstmpt 5738 . . . . . . . . . . 11 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
1413cnveqi 5874 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
15 cnvxp 6156 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = ({𝐴} × 𝑆)
1614, 15eqtr3i 2762 . . . . . . . . 9 (𝑥 𝑆𝐴) = ({𝐴} × 𝑆)
1716imaeq1i 6056 . . . . . . . 8 ((𝑥 𝑆𝐴) “ 𝑦) = (({𝐴} × 𝑆) “ 𝑦)
18 df-ima 5689 . . . . . . . 8 (({𝐴} × 𝑆) “ 𝑦) = ran (({𝐴} × 𝑆) ↾ 𝑦)
19 df-rn 5687 . . . . . . . 8 ran (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
2017, 18, 193eqtri 2764 . . . . . . 7 ((𝑥 𝑆𝐴) “ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
21 df-res 5688 . . . . . . . . . 10 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} × 𝑆) ∩ (𝑦 × V))
22 inxp 5832 . . . . . . . . . 10 (({𝐴} × 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V))
23 inv1 4394 . . . . . . . . . . 11 ( 𝑆 ∩ V) = 𝑆
2423xpeq2i 5703 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × 𝑆)
2521, 22, 243eqtri 2764 . . . . . . . . 9 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2625cnveqi 5874 . . . . . . . 8 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2726dmeqi 5904 . . . . . . 7 dom (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} ∩ 𝑦) × 𝑆)
28 cnvxp 6156 . . . . . . . 8 (({𝐴} ∩ 𝑦) × 𝑆) = ( 𝑆 × ({𝐴} ∩ 𝑦))
2928dmeqi 5904 . . . . . . 7 dom (({𝐴} ∩ 𝑦) × 𝑆) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
3020, 27, 293eqtri 2764 . . . . . 6 ((𝑥 𝑆𝐴) “ 𝑦) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
31 xpeq2 5697 . . . . . . . . . . 11 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ( 𝑆 × ∅))
32 xp0 6157 . . . . . . . . . . 11 ( 𝑆 × ∅) = ∅
3331, 32eqtrdi 2788 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3433dmeqd 5905 . . . . . . . . 9 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35 dm0 5920 . . . . . . . . 9 dom ∅ = ∅
3634, 35eqtrdi 2788 . . . . . . . 8 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3736adantl 482 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38 0elsiga 33181 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
398, 38syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑆)
4039adantr 481 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
4137, 40eqeltrd 2833 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4230, 41eqeltrid 2837 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
43 dmxp 5928 . . . . . . . 8 (({𝐴} ∩ 𝑦) ≠ ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4443adantl 482 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4510adantr 481 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → 𝑆𝑆)
4644, 45eqeltrd 2833 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4730, 46eqeltrid 2837 . . . . 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 5875 . . . . . 6 (𝜑𝐹 = (𝑥 𝑆𝐴))
5150imaeq1d 6058 . . . . 5 (𝜑 → (𝐹𝑦) = ((𝑥 𝑆𝐴) “ 𝑦))
5251eleq1d 2818 . . . 4 (𝜑 → ((𝐹𝑦) ∈ 𝑆 ↔ ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5352ralbidv 3177 . . 3 (𝜑 → (∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5449, 53mpbird 256 . 2 (𝜑 → ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)
558, 5ismbfm 33318 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)))
5612, 54, 55mpbir2and 711 1 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  Vcvv 3474  cin 3947  c0 4322  {csn 4628   cuni 4908  cmpt 5231   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  wf 6539  (class class class)co 7411  m cmap 8822  sigAlgebracsiga 33175  MblFnMcmbfm 33316
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-siga 33176  df-mbfm 33317
This theorem is referenced by:  sibf0  33402
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