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Theorem ismbfm 34252
Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 25663. (Contributed by Thierry Arnoux, 23-Jan-2017.)
Hypotheses
Ref Expression
ismbfm.1 (𝜑𝑆 ran sigAlgebra)
ismbfm.2 (𝜑𝑇 ran sigAlgebra)
Assertion
Ref Expression
ismbfm (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑆   𝑥,𝑇
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ismbfm
Dummy variables 𝑓 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismbfm.1 . . . 4 (𝜑𝑆 ran sigAlgebra)
2 ismbfm.2 . . . 4 (𝜑𝑇 ran sigAlgebra)
3 unieq 4918 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 7447 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡m 𝑠) = ( 𝑡m 𝑆))
5 eleq2 2830 . . . . . . 7 (𝑠 = 𝑆 → ((𝑓𝑥) ∈ 𝑠 ↔ (𝑓𝑥) ∈ 𝑆))
65ralbidv 3178 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆))
74, 6rabeqbidv 3455 . . . . 5 (𝑠 = 𝑆 → {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} = {𝑓 ∈ ( 𝑡m 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆})
8 unieq 4918 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
98oveq1d 7446 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡m 𝑆) = ( 𝑇m 𝑆))
10 raleq 3323 . . . . . 6 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆))
119, 10rabeqbidv 3455 . . . . 5 (𝑡 = 𝑇 → {𝑓 ∈ ( 𝑡m 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆} = {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
12 df-mbfm 34251 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
13 ovex 7464 . . . . . 6 ( 𝑇m 𝑆) ∈ V
1413rabex 5339 . . . . 5 {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ∈ V
157, 11, 12, 14ovmpo 7593 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
161, 2, 15syl2anc 584 . . 3 (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
1716eleq2d 2827 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆}))
18 cnveq 5884 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
1918imaeq1d 6077 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
2019eleq1d 2826 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2120ralbidv 3178 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2221elrab 3692 . 2 (𝐹 ∈ {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2317, 22bitrdi 287 1 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436   cuni 4907  ccnv 5684  ran crn 5686  cima 5688  (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-pr 5432
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-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-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-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-mbfm 34251
This theorem is referenced by:  elunirnmbfm  34253  mbfmf  34255  mbfmcnvima  34257  mbfmcst  34261  1stmbfm  34262  2ndmbfm  34263  imambfm  34264  mbfmco  34266  elmbfmvol2  34269  mbfmcnt  34270  sibfof  34342  isrrvv  34445
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