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Theorem ismbfm 30644
Description: The predicate "𝐹 is a measurable function from the measurable space 𝑆 to the measurable space 𝑇". Cf. ismbf 23616. (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 4596 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
43oveq2d 6830 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡𝑚 𝑠) = ( 𝑡𝑚 𝑆))
5 eleq2 2828 . . . . . . 7 (𝑠 = 𝑆 → ((𝑓𝑥) ∈ 𝑠 ↔ (𝑓𝑥) ∈ 𝑆))
65ralbidv 3124 . . . . . 6 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆))
74, 6rabeqbidv 3335 . . . . 5 (𝑠 = 𝑆 → {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠} = {𝑓 ∈ ( 𝑡𝑚 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆})
8 unieq 4596 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
98oveq1d 6829 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡𝑚 𝑆) = ( 𝑇𝑚 𝑆))
10 raleq 3277 . . . . . 6 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆))
119, 10rabeqbidv 3335 . . . . 5 (𝑡 = 𝑇 → {𝑓 ∈ ( 𝑡𝑚 𝑆) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑆} = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
12 df-mbfm 30643 . . . . 5 MblFnM = (𝑠 ran sigAlgebra, 𝑡 ran sigAlgebra ↦ {𝑓 ∈ ( 𝑡𝑚 𝑠) ∣ ∀𝑥𝑡 (𝑓𝑥) ∈ 𝑠})
13 ovex 6842 . . . . . 6 ( 𝑇𝑚 𝑆) ∈ V
1413rabex 4964 . . . . 5 {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆} ∈ V
157, 11, 12, 14ovmpt2 6962 . . . 4 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra) → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
161, 2, 15syl2anc 696 . . 3 (𝜑 → (𝑆MblFnM𝑇) = {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆})
1716eleq2d 2825 . 2 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ 𝐹 ∈ {𝑓 ∈ ( 𝑇𝑚 𝑆) ∣ ∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆}))
18 cnveq 5451 . . . . . 6 (𝑓 = 𝐹𝑓 = 𝐹)
1918imaeq1d 5623 . . . . 5 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
2019eleq1d 2824 . . . 4 (𝑓 = 𝐹 → ((𝑓𝑥) ∈ 𝑆 ↔ (𝐹𝑥) ∈ 𝑆))
2120ralbidv 3124 . . 3 (𝑓 = 𝐹 → (∀𝑥𝑇 (𝑓𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
2221elrab 3504 . 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 383   = wceq 1632  wcel 2139  wral 3050  {crab 3054   cuni 4588  ccnv 5265  ran crn 5267  cima 5269  (class class class)co 6814  𝑚 cmap 8025  sigAlgebracsiga 30500  MblFnMcmbfm 30642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-mbfm 30643
This theorem is referenced by:  elunirnmbfm  30645  mbfmf  30647  isanmbfm  30648  mbfmcnvima  30649  mbfmcst  30651  1stmbfm  30652  2ndmbfm  30653  imambfm  30654  mbfmco  30656  elmbfmvol2  30659  mbfmcnt  30660  sibfof  30732  isrrvv  30835
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