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Theorem mbfmcst 30551
Description: A constant function is measurable. Cf. mbfconst 23522. (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 472 . . . 4 ((𝜑𝑥 𝑆) → 𝐴 𝑇)
41, 3fmpt3d 6501 . . 3 (𝜑𝐹: 𝑆 𝑇)
5 mbfmcst.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
6 unielsiga 30421 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
75, 6syl 17 . . . 4 (𝜑 𝑇𝑇)
8 mbfmcst.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
9 unielsiga 30421 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
108, 9syl 17 . . . 4 (𝜑 𝑆𝑆)
117, 10elmapd 7988 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇𝑚 𝑆) ↔ 𝐹: 𝑆 𝑇))
124, 11mpbird 247 . 2 (𝜑𝐹 ∈ ( 𝑇𝑚 𝑆))
13 fconstmpt 5272 . . . . . . . . . . 11 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
1413cnveqi 5404 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
15 cnvxp 5661 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = ({𝐴} × 𝑆)
1614, 15eqtr3i 2748 . . . . . . . . 9 (𝑥 𝑆𝐴) = ({𝐴} × 𝑆)
1716imaeq1i 5573 . . . . . . . 8 ((𝑥 𝑆𝐴) “ 𝑦) = (({𝐴} × 𝑆) “ 𝑦)
18 df-ima 5231 . . . . . . . 8 (({𝐴} × 𝑆) “ 𝑦) = ran (({𝐴} × 𝑆) ↾ 𝑦)
19 df-rn 5229 . . . . . . . 8 ran (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
2017, 18, 193eqtri 2750 . . . . . . 7 ((𝑥 𝑆𝐴) “ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
21 df-res 5230 . . . . . . . . . 10 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} × 𝑆) ∩ (𝑦 × V))
22 inxp 5362 . . . . . . . . . 10 (({𝐴} × 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V))
23 inv1 4078 . . . . . . . . . . 11 ( 𝑆 ∩ V) = 𝑆
2423xpeq2i 5245 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × 𝑆)
2521, 22, 243eqtri 2750 . . . . . . . . 9 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2625cnveqi 5404 . . . . . . . 8 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2726dmeqi 5432 . . . . . . 7 dom (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} ∩ 𝑦) × 𝑆)
28 cnvxp 5661 . . . . . . . 8 (({𝐴} ∩ 𝑦) × 𝑆) = ( 𝑆 × ({𝐴} ∩ 𝑦))
2928dmeqi 5432 . . . . . . 7 dom (({𝐴} ∩ 𝑦) × 𝑆) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
3020, 27, 293eqtri 2750 . . . . . 6 ((𝑥 𝑆𝐴) “ 𝑦) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
31 xpeq2 5238 . . . . . . . . . . 11 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ( 𝑆 × ∅))
32 xp0 5662 . . . . . . . . . . 11 ( 𝑆 × ∅) = ∅
3331, 32syl6eq 2774 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3433dmeqd 5433 . . . . . . . . 9 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35 dm0 5446 . . . . . . . . 9 dom ∅ = ∅
3634, 35syl6eq 2774 . . . . . . . 8 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3736adantl 473 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38 0elsiga 30407 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
398, 38syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑆)
4039adantr 472 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
4137, 40eqeltrd 2803 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4230, 41syl5eqel 2807 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
43 dmxp 5451 . . . . . . . 8 (({𝐴} ∩ 𝑦) ≠ ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4443adantl 473 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4510adantr 472 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → 𝑆𝑆)
4644, 45eqeltrd 2803 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4730, 46syl5eqel 2807 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4842, 47pm2.61dane 2983 . . . 4 (𝜑 → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4948ralrimivw 3069 . . 3 (𝜑 → ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
501cnveqd 5405 . . . . . 6 (𝜑𝐹 = (𝑥 𝑆𝐴))
5150imaeq1d 5575 . . . . 5 (𝜑 → (𝐹𝑦) = ((𝑥 𝑆𝐴) “ 𝑦))
5251eleq1d 2788 . . . 4 (𝜑 → ((𝐹𝑦) ∈ 𝑆 ↔ ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5352ralbidv 3088 . . 3 (𝜑 → (∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5449, 53mpbird 247 . 2 (𝜑 → ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)
558, 5ismbfm 30544 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)))
5612, 54, 55mpbir2and 995 1 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wne 2896  wral 3014  Vcvv 3304  cin 3679  c0 4023  {csn 4285   cuni 4544  cmpt 4837   × cxp 5216  ccnv 5217  dom cdm 5218  ran crn 5219  cres 5220  cima 5221  wf 5997  (class class class)co 6765  𝑚 cmap 7974  sigAlgebracsiga 30400  MblFnMcmbfm 30542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-fal 1602  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-map 7976  df-siga 30401  df-mbfm 30543
This theorem is referenced by:  sibf0  30626
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