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Theorem mbfmcst 34240
Description: A constant function is measurable. Cf. mbfconst 25681. (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 7135 . . 3 (𝜑𝐹: 𝑆 𝑇)
5 mbfmcst.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
6 unielsiga 34108 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
75, 6syl 17 . . . 4 (𝜑 𝑇𝑇)
8 mbfmcst.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
9 unielsiga 34108 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
108, 9syl 17 . . . 4 (𝜑 𝑆𝑆)
117, 10elmapd 8878 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ↔ 𝐹: 𝑆 𝑇))
124, 11mpbird 257 . 2 (𝜑𝐹 ∈ ( 𝑇m 𝑆))
13 fconstmpt 5750 . . . . . . . . . . 11 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
1413cnveqi 5887 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
15 cnvxp 6178 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = ({𝐴} × 𝑆)
1614, 15eqtr3i 2764 . . . . . . . . 9 (𝑥 𝑆𝐴) = ({𝐴} × 𝑆)
1716imaeq1i 6076 . . . . . . . 8 ((𝑥 𝑆𝐴) “ 𝑦) = (({𝐴} × 𝑆) “ 𝑦)
18 df-ima 5701 . . . . . . . 8 (({𝐴} × 𝑆) “ 𝑦) = ran (({𝐴} × 𝑆) ↾ 𝑦)
19 df-rn 5699 . . . . . . . 8 ran (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
2017, 18, 193eqtri 2766 . . . . . . 7 ((𝑥 𝑆𝐴) “ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
21 df-res 5700 . . . . . . . . . 10 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} × 𝑆) ∩ (𝑦 × V))
22 inxp 5844 . . . . . . . . . 10 (({𝐴} × 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V))
23 inv1 4403 . . . . . . . . . . 11 ( 𝑆 ∩ V) = 𝑆
2423xpeq2i 5715 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × 𝑆)
2521, 22, 243eqtri 2766 . . . . . . . . 9 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2625cnveqi 5887 . . . . . . . 8 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2726dmeqi 5917 . . . . . . 7 dom (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} ∩ 𝑦) × 𝑆)
28 cnvxp 6178 . . . . . . . 8 (({𝐴} ∩ 𝑦) × 𝑆) = ( 𝑆 × ({𝐴} ∩ 𝑦))
2928dmeqi 5917 . . . . . . 7 dom (({𝐴} ∩ 𝑦) × 𝑆) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
3020, 27, 293eqtri 2766 . . . . . 6 ((𝑥 𝑆𝐴) “ 𝑦) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
31 xpeq2 5709 . . . . . . . . . . 11 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ( 𝑆 × ∅))
32 xp0 6179 . . . . . . . . . . 11 ( 𝑆 × ∅) = ∅
3331, 32eqtrdi 2790 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3433dmeqd 5918 . . . . . . . . 9 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35 dm0 5933 . . . . . . . . 9 dom ∅ = ∅
3634, 35eqtrdi 2790 . . . . . . . 8 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3736adantl 481 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38 0elsiga 34094 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
398, 38syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑆)
4039adantr 480 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
4137, 40eqeltrd 2838 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4230, 41eqeltrid 2842 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
43 dmxp 5941 . . . . . . . 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 2838 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4730, 46eqeltrid 2842 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4842, 47pm2.61dane 3026 . . . 4 (𝜑 → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4948ralrimivw 3147 . . 3 (𝜑 → ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
501cnveqd 5888 . . . . . 6 (𝜑𝐹 = (𝑥 𝑆𝐴))
5150imaeq1d 6078 . . . . 5 (𝜑 → (𝐹𝑦) = ((𝑥 𝑆𝐴) “ 𝑦))
5251eleq1d 2823 . . . 4 (𝜑 → ((𝐹𝑦) ∈ 𝑆 ↔ ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5352ralbidv 3175 . . 3 (𝜑 → (∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5449, 53mpbird 257 . 2 (𝜑 → ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)
558, 5ismbfm 34231 . 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 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  cin 3961  c0 4338  {csn 4630   cuni 4911  cmpt 5230   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  cima 5691  wf 6558  (class class class)co 7430  m cmap 8864  sigAlgebracsiga 34088  MblFnMcmbfm 34229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-siga 34089  df-mbfm 34230
This theorem is referenced by:  sibf0  34315
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