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Theorem mbfmcst 31625
Description: A constant function is measurable. Cf. mbfconst 24240. (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 484 . . . 4 ((𝜑𝑥 𝑆) → 𝐴 𝑇)
41, 3fmpt3d 6861 . . 3 (𝜑𝐹: 𝑆 𝑇)
5 mbfmcst.2 . . . . 5 (𝜑𝑇 ran sigAlgebra)
6 unielsiga 31495 . . . . 5 (𝑇 ran sigAlgebra → 𝑇𝑇)
75, 6syl 17 . . . 4 (𝜑 𝑇𝑇)
8 mbfmcst.1 . . . . 5 (𝜑𝑆 ran sigAlgebra)
9 unielsiga 31495 . . . . 5 (𝑆 ran sigAlgebra → 𝑆𝑆)
108, 9syl 17 . . . 4 (𝜑 𝑆𝑆)
117, 10elmapd 8407 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ↔ 𝐹: 𝑆 𝑇))
124, 11mpbird 260 . 2 (𝜑𝐹 ∈ ( 𝑇m 𝑆))
13 fconstmpt 5582 . . . . . . . . . . 11 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
1413cnveqi 5713 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = (𝑥 𝑆𝐴)
15 cnvxp 5985 . . . . . . . . . 10 ( 𝑆 × {𝐴}) = ({𝐴} × 𝑆)
1614, 15eqtr3i 2826 . . . . . . . . 9 (𝑥 𝑆𝐴) = ({𝐴} × 𝑆)
1716imaeq1i 5897 . . . . . . . 8 ((𝑥 𝑆𝐴) “ 𝑦) = (({𝐴} × 𝑆) “ 𝑦)
18 df-ima 5536 . . . . . . . 8 (({𝐴} × 𝑆) “ 𝑦) = ran (({𝐴} × 𝑆) ↾ 𝑦)
19 df-rn 5534 . . . . . . . 8 ran (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
2017, 18, 193eqtri 2828 . . . . . . 7 ((𝑥 𝑆𝐴) “ 𝑦) = dom (({𝐴} × 𝑆) ↾ 𝑦)
21 df-res 5535 . . . . . . . . . 10 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} × 𝑆) ∩ (𝑦 × V))
22 inxp 5671 . . . . . . . . . 10 (({𝐴} × 𝑆) ∩ (𝑦 × V)) = (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V))
23 inv1 4305 . . . . . . . . . . 11 ( 𝑆 ∩ V) = 𝑆
2423xpeq2i 5550 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) × ( 𝑆 ∩ V)) = (({𝐴} ∩ 𝑦) × 𝑆)
2521, 22, 243eqtri 2828 . . . . . . . . 9 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2625cnveqi 5713 . . . . . . . 8 (({𝐴} × 𝑆) ↾ 𝑦) = (({𝐴} ∩ 𝑦) × 𝑆)
2726dmeqi 5741 . . . . . . 7 dom (({𝐴} × 𝑆) ↾ 𝑦) = dom (({𝐴} ∩ 𝑦) × 𝑆)
28 cnvxp 5985 . . . . . . . 8 (({𝐴} ∩ 𝑦) × 𝑆) = ( 𝑆 × ({𝐴} ∩ 𝑦))
2928dmeqi 5741 . . . . . . 7 dom (({𝐴} ∩ 𝑦) × 𝑆) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
3020, 27, 293eqtri 2828 . . . . . 6 ((𝑥 𝑆𝐴) “ 𝑦) = dom ( 𝑆 × ({𝐴} ∩ 𝑦))
31 xpeq2 5544 . . . . . . . . . . 11 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ( 𝑆 × ∅))
32 xp0 5986 . . . . . . . . . . 11 ( 𝑆 × ∅) = ∅
3331, 32eqtrdi 2852 . . . . . . . . . 10 (({𝐴} ∩ 𝑦) = ∅ → ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3433dmeqd 5742 . . . . . . . . 9 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = dom ∅)
35 dm0 5758 . . . . . . . . 9 dom ∅ = ∅
3634, 35eqtrdi 2852 . . . . . . . 8 (({𝐴} ∩ 𝑦) = ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
3736adantl 485 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = ∅)
38 0elsiga 31481 . . . . . . . . 9 (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
398, 38syl 17 . . . . . . . 8 (𝜑 → ∅ ∈ 𝑆)
4039adantr 484 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ∅ ∈ 𝑆)
4137, 40eqeltrd 2893 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4230, 41eqeltrid 2897 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) = ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
43 dmxp 5767 . . . . . . . 8 (({𝐴} ∩ 𝑦) ≠ ∅ → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4443adantl 485 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) = 𝑆)
4510adantr 484 . . . . . . 7 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → 𝑆𝑆)
4644, 45eqeltrd 2893 . . . . . 6 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → dom ( 𝑆 × ({𝐴} ∩ 𝑦)) ∈ 𝑆)
4730, 46eqeltrid 2897 . . . . 5 ((𝜑 ∧ ({𝐴} ∩ 𝑦) ≠ ∅) → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4842, 47pm2.61dane 3077 . . . 4 (𝜑 → ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
4948ralrimivw 3153 . . 3 (𝜑 → ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆)
501cnveqd 5714 . . . . . 6 (𝜑𝐹 = (𝑥 𝑆𝐴))
5150imaeq1d 5899 . . . . 5 (𝜑 → (𝐹𝑦) = ((𝑥 𝑆𝐴) “ 𝑦))
5251eleq1d 2877 . . . 4 (𝜑 → ((𝐹𝑦) ∈ 𝑆 ↔ ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5352ralbidv 3165 . . 3 (𝜑 → (∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆 ↔ ∀𝑦𝑇 ((𝑥 𝑆𝐴) “ 𝑦) ∈ 𝑆))
5449, 53mpbird 260 . 2 (𝜑 → ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)
558, 5ismbfm 31618 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑦𝑇 (𝐹𝑦) ∈ 𝑆)))
5612, 54, 55mpbir2and 712 1 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  cin 3883  c0 4246  {csn 4528   cuni 4803  cmpt 5113   × cxp 5521  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  wf 6324  (class class class)co 7139  m cmap 8393  sigAlgebracsiga 31475  MblFnMcmbfm 31616
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-siga 31476  df-mbfm 31617
This theorem is referenced by:  sibf0  31700
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