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Theorem mbfmfun 28688
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 28687 . . 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 elmapfun 7479 . . . . 5  |-  ( F  e.  ( U. t  ^m  U. s )  ->  Fun  F )
54adantr 463 . . . 4  |-  ( ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s )  ->  Fun  F )
65rexlimivw 2892 . . 3  |-  ( E. t  e.  U. ran sigAlgebra ( F  e.  ( U. t  ^m  U. s )  /\  A. x  e.  t  ( `' F " x )  e.  s )  ->  Fun  F )
76rexlimivw 2892 . 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 )
81, 3, 73syl 20 1  |-  ( ph  ->  Fun  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    e. wcel 1842   A.wral 2753   E.wrex 2754   U.cuni 4190   `'ccnv 4821   ran crn 4823   "cima 4825   Fun wfun 5562  (class class class)co 6277    ^m cmap 7456  sigAlgebracsiga 28541  MblFnMcmbfm 28684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-map 7458  df-mbfm 28685
This theorem is referenced by:  orvcval4  28891
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