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Theorem isanmbfm 31123
Description: The predicate to be a measurable function. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypotheses
Ref Expression
mbfmf.1 (𝜑𝑆 ran sigAlgebra)
mbfmf.2 (𝜑𝑇 ran sigAlgebra)
mbfmf.3 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
Assertion
Ref Expression
isanmbfm (𝜑𝐹 ran MblFnM)

Proof of Theorem isanmbfm
Dummy variables 𝑡 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfmf.1 . . 3 (𝜑𝑆 ran sigAlgebra)
2 mbfmf.2 . . 3 (𝜑𝑇 ran sigAlgebra)
3 mbfmf.3 . . . 4 (𝜑𝐹 ∈ (𝑆MblFnM𝑇))
41, 2ismbfm 31119 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
53, 4mpbid 233 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
6 unieq 4755 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
76oveq2d 7035 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡𝑚 𝑠) = ( 𝑡𝑚 𝑆))
87eleq2d 2867 . . . . 5 (𝑠 = 𝑆 → (𝐹 ∈ ( 𝑡𝑚 𝑠) ↔ 𝐹 ∈ ( 𝑡𝑚 𝑆)))
9 eleq2 2870 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑥) ∈ 𝑠 ↔ (𝐹𝑥) ∈ 𝑆))
109ralbidv 3163 . . . . 5 (𝑠 = 𝑆 → (∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠 ↔ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆))
118, 10anbi12d 630 . . . 4 (𝑠 = 𝑆 → ((𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) ↔ (𝐹 ∈ ( 𝑡𝑚 𝑆) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆)))
12 unieq 4755 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
1312oveq1d 7034 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡𝑚 𝑆) = ( 𝑇𝑚 𝑆))
1413eleq2d 2867 . . . . 5 (𝑡 = 𝑇 → (𝐹 ∈ ( 𝑡𝑚 𝑆) ↔ 𝐹 ∈ ( 𝑇𝑚 𝑆)))
15 raleq 3364 . . . . 5 (𝑡 = 𝑇 → (∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆 ↔ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
1614, 15anbi12d 630 . . . 4 (𝑡 = 𝑇 → ((𝐹 ∈ ( 𝑡𝑚 𝑆) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑆) ↔ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
1711, 16rspc2ev 3572 . . 3 ((𝑆 ran sigAlgebra ∧ 𝑇 ran sigAlgebra ∧ (𝐹 ∈ ( 𝑇𝑚 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)) → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
181, 2, 5, 17syl3anc 1364 . 2 (𝜑 → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
19 elunirnmbfm 31120 . 2 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡𝑚 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
2018, 19sylibr 235 1 (𝜑𝐹 ran MblFnM)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2080  wral 3104  wrex 3105   cuni 4747  ccnv 5445  ran crn 5447  cima 5449  (class class class)co 7019  𝑚 cmap 8259  sigAlgebracsiga 30976  MblFnMcmbfm 31117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ne 2984  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-csb 3814  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-iun 4829  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-res 5458  df-ima 5459  df-iota 6192  df-fun 6230  df-fn 6231  df-f 6232  df-fv 6236  df-ov 7022  df-oprab 7023  df-mpo 7024  df-1st 7548  df-2nd 7549  df-mbfm 31118
This theorem is referenced by:  mbfmbfm  31125  orvcval4  31327
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