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Theorem isanmbfm 31413
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 31409 . . . 4 (𝜑 → (𝐹 ∈ (𝑆MblFnM𝑇) ↔ (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆)))
53, 4mpbid 233 . . 3 (𝜑 → (𝐹 ∈ ( 𝑇m 𝑆) ∧ ∀𝑥𝑇 (𝐹𝑥) ∈ 𝑆))
6 unieq 4838 . . . . . . 7 (𝑠 = 𝑆 𝑠 = 𝑆)
76oveq2d 7161 . . . . . 6 (𝑠 = 𝑆 → ( 𝑡m 𝑠) = ( 𝑡m 𝑆))
87eleq2d 2895 . . . . 5 (𝑠 = 𝑆 → (𝐹 ∈ ( 𝑡m 𝑠) ↔ 𝐹 ∈ ( 𝑡m 𝑆)))
9 eleq2 2898 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑥) ∈ 𝑠 ↔ (𝐹𝑥) ∈ 𝑆))
109ralbidv 3194 . . . . 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 4838 . . . . . . 7 (𝑡 = 𝑇 𝑡 = 𝑇)
1312oveq1d 7160 . . . . . 6 (𝑡 = 𝑇 → ( 𝑡m 𝑆) = ( 𝑇m 𝑆))
1413eleq2d 2895 . . . . 5 (𝑡 = 𝑇 → (𝐹 ∈ ( 𝑡m 𝑆) ↔ 𝐹 ∈ ( 𝑇m 𝑆)))
15 raleq 3403 . . . . 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 3632 . . 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 1363 . 2 (𝜑 → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
19 elunirnmbfm 31410 . 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 1528  wcel 2105  wral 3135  wrex 3136   cuni 4830  ccnv 5547  ran crn 5549  cima 5551  (class class class)co 7145  m cmap 8395  sigAlgebracsiga 31266  MblFnMcmbfm 31407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-mbfm 31408
This theorem is referenced by:  mbfmbfm  31415  orvcval4  31617
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