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Theorem mbfmfun 34234
Description: A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Hypothesis
Ref Expression
mbfmfun.1 (𝜑𝐹 ran MblFnM)
Assertion
Ref Expression
mbfmfun (𝜑 → Fun 𝐹)

Proof of Theorem mbfmfun
Dummy variables 𝑡 𝑠 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mbfmfun.1 . 2 (𝜑𝐹 ran MblFnM)
2 elunirnmbfm 34233 . . 3 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
32biimpi 216 . 2 (𝐹 ran MblFnM → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
4 elmapfun 8905 . . . . 5 (𝐹 ∈ ( 𝑡m 𝑠) → Fun 𝐹)
54adantr 480 . . . 4 ((𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
65rexlimivw 3149 . . 3 (∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
76rexlimivw 3149 . 2 (∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
81, 3, 73syl 18 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2106  wral 3059  wrex 3068   cuni 4912  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557  (class class class)co 7431  m cmap 8865  sigAlgebracsiga 34089  MblFnMcmbfm 34230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-mbfm 34231
This theorem is referenced by:  orvcval4  34442
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