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Theorem ismbfm 30764
Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23689. (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 4604 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 6860 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡𝑚 𝑠) = ( 𝑡𝑚 𝑆))
5 eleq2 2833 . . . . . . 7 (𝑠 = 𝑆 → ((𝑓𝑥) ∈ 𝑠 ↔ (𝑓𝑥) ∈ 𝑆))
65ralbidv 3133 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆))
74, 6rabeqbidv 3344 . . . . 5 (𝑠 = 𝑆 → {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} = {𝑓 ∈ ( 𝑡𝑚 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆})
8 unieq 4604 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
98oveq1d 6859 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡𝑚 𝑆) = ( 𝑇𝑚 𝑆))
10 raleq 3286 . . . . . 6 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆))
119, 10rabeqbidv 3344 . . . . 5 (𝑡 = 𝑇 → {𝑓 ∈ ( 𝑡𝑚 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆} = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
12 df-mbfm 30763 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
13 ovex 6876 . . . . . 6 ( 𝑇𝑚 𝑆) ∈ V
1413rabex 4975 . . . . 5 {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ∈ V
157, 11, 12, 14ovmpt2 6996 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
161, 2, 15syl2anc 579 . . 3 (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
1716eleq2d 2830 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆}))
18 cnveq 5466 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
1918imaeq1d 5649 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
2019eleq1d 2829 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2120ralbidv 3133 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2221elrab 3521 . 2 (𝐹 ∈ {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2317, 22syl6bb 278 1 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059   cuni 4596  ccnv 5278  ran crn 5280  cima 5282  (class class class)co 6844  𝑚 cmap 8062  sigAlgebracsiga 30620  MblFnMcmbfm 30762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-iota 6033  df-fun 6072  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-mbfm 30763
This theorem is referenced by:  elunirnmbfm  30765  mbfmf  30767  isanmbfm  30768  mbfmcnvima  30769  mbfmcst  30771  1stmbfm  30772  2ndmbfm  30773  imambfm  30774  mbfmco  30776  elmbfmvol2  30779  mbfmcnt  30780  sibfof  30852  isrrvv  30956
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