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Theorem ismbfm 30095
Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23303. (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 4410 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 6620 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡𝑚 𝑠) = ( 𝑡𝑚 𝑆))
5 eleq2 2687 . . . . . . 7 (𝑠 = 𝑆 → ((𝑓𝑥) ∈ 𝑠 ↔ (𝑓𝑥) ∈ 𝑆))
65ralbidv 2980 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆))
74, 6rabeqbidv 3181 . . . . 5 (𝑠 = 𝑆 → {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} = {𝑓 ∈ ( 𝑡𝑚 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆})
8 unieq 4410 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
98oveq1d 6619 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡𝑚 𝑆) = ( 𝑇𝑚 𝑆))
10 raleq 3127 . . . . . 6 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆))
119, 10rabeqbidv 3181 . . . . 5 (𝑡 = 𝑇 → {𝑓 ∈ ( 𝑡𝑚 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆} = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
12 df-mbfm 30094 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
13 ovex 6632 . . . . . 6 ( 𝑇𝑚 𝑆) ∈ V
1413rabex 4773 . . . . 5 {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ∈ V
157, 11, 12, 14ovmpt2 6749 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
161, 2, 15syl2anc 692 . . 3 (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
1716eleq2d 2684 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆}))
18 cnveq 5256 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
1918imaeq1d 5424 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
2019eleq1d 2683 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2120ralbidv 2980 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2221elrab 3346 . 2 (𝐹 ∈ {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2317, 22syl6bb 276 1 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911   cuni 4402  ccnv 5073  ran crn 5075  cima 5077  (class class class)co 6604  𝑚 cmap 7802  sigAlgebracsiga 29951  MblFnMcmbfm 30093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-mbfm 30094
This theorem is referenced by:  elunirnmbfm  30096  mbfmf  30098  isanmbfm  30099  mbfmcnvima  30100  mbfmcst  30102  1stmbfm  30103  2ndmbfm  30104  imambfm  30105  mbfmco  30107  elmbfmvol2  30110  mbfmcnt  30111  sibfof  30183  isrrvv  30286
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