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Theorem ismbfm 32219
Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 24792. (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 4850 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 7291 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡m 𝑠) = ( 𝑡m 𝑆))
5 eleq2 2827 . . . . . . 7 (𝑠 = 𝑆 → ((𝑓𝑥) ∈ 𝑠 ↔ (𝑓𝑥) ∈ 𝑆))
65ralbidv 3112 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆))
74, 6rabeqbidv 3420 . . . . 5 (𝑠 = 𝑆 → {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} = {𝑓 ∈ ( 𝑡m 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆})
8 unieq 4850 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
98oveq1d 7290 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡m 𝑆) = ( 𝑇m 𝑆))
10 raleq 3342 . . . . . 6 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆))
119, 10rabeqbidv 3420 . . . . 5 (𝑡 = 𝑇 → {𝑓 ∈ ( 𝑡m 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆} = {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
12 df-mbfm 32218 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡m 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
13 ovex 7308 . . . . . 6 ( 𝑇m 𝑆) ∈ V
1413rabex 5256 . . . . 5 {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ∈ V
157, 11, 12, 14ovmpo 7433 . . . 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 2824 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ ( 𝑇m 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆}))
18 cnveq 5782 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
1918imaeq1d 5968 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
2019eleq1d 2823 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2120ralbidv 3112 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2221elrab 3624 . 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 205  wa 396   = wceq 1539  wcel 2106  wral 3064  {crab 3068   cuni 4839  ccnv 5588  ran crn 5590  cima 5592  (class class class)co 7275  m cmap 8615  sigAlgebracsiga 32076  MblFnMcmbfm 32217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-mbfm 32218
This theorem is referenced by:  elunirnmbfm  32220  mbfmf  32222  isanmbfm  32223  mbfmcnvima  32224  mbfmcst  32226  1stmbfm  32227  2ndmbfm  32228  imambfm  32229  mbfmco  32231  elmbfmvol2  32234  mbfmcnt  32235  sibfof  32307  isrrvv  32410
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