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Theorem mbfmfun 31539
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 31538 . . 3 (𝐹 ran MblFnM ↔ ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
32biimpi 219 . 2 (𝐹 ran MblFnM → ∃𝑠 ran sigAlgebra∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠))
4 elmapfun 8422 . . . . 5 (𝐹 ∈ ( 𝑡m 𝑠) → Fun 𝐹)
54adantr 484 . . . 4 ((𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
65rexlimivw 3275 . . 3 (∃𝑡 ran sigAlgebra(𝐹 ∈ ( 𝑡m 𝑠) ∧ ∀𝑥𝑡 (𝐹𝑥) ∈ 𝑠) → Fun 𝐹)
76rexlimivw 3275 . 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 399  wcel 2115  wral 3133  wrex 3134   cuni 4825  ccnv 5542  ran crn 5544  cima 5546  Fun wfun 6338  (class class class)co 7146  m cmap 8398  sigAlgebracsiga 31394  MblFnMcmbfm 31535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-1st 7681  df-2nd 7682  df-map 8400  df-mbfm 31536
This theorem is referenced by:  orvcval4  31745
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