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Theorem mbfmfun 27851
Description: A measurable function is a function. (Contributed by Thierry Arnoux, 24-Jan-2017.)
Hypothesis
Ref Expression
mbfmfun.1  |-  ( ph  ->  F  e.  U. ran MblFnM )
Assertion
Ref Expression
mbfmfun  |-  ( ph  ->  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  |-  ( ph  ->  F  e.  U. ran MblFnM )
2 elunirnmbfm 27850 . . 3  |-  ( F  e.  U. ran MblFnM  <->  E. s  e.  U. ran sigAlgebra E. t  e. 
U. ran sigAlgebra ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s ) )
32biimpi 194 . 2  |-  ( F  e.  U. ran MblFnM  ->  E. s  e.  U. ran sigAlgebra E. t  e. 
U. ran sigAlgebra ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s ) )
4 elmapi 7430 . . . . . 6  |-  ( F  e.  ( U. t  ^m  U. s )  ->  F : U. s --> U. t )
5 ffun 5724 . . . . . 6  |-  ( F : U. s --> U. t  ->  Fun  F )
64, 5syl 16 . . . . 5  |-  ( F  e.  ( U. t  ^m  U. s )  ->  Fun  F )
76adantr 465 . . . 4  |-  ( ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s )  ->  Fun  F )
87rexlimivw 2945 . . 3  |-  ( E. t  e.  U. ran sigAlgebra ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s )  ->  Fun  F )
98rexlimivw 2945 . 2  |-  ( E. s  e.  U. ran sigAlgebra E. t  e.  U. ran sigAlgebra ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s )  ->  Fun  F )
101, 3, 93syl 20 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   A.wral 2807   E.wrex 2808   U.cuni 4238   `'ccnv 4991   ran crn 4993   "cima 4995   Fun wfun 5573   -->wf 5575  (class class class)co 6275    ^m cmap 7410  sigAlgebracsiga 27733  MblFnMcmbfm 27847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-map 7412  df-mbfm 27848
This theorem is referenced by:  orvcval4  28025
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