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Theorem orvcoel 29303
Description: If the relation produces open sets, preimage maps by a measurable function are measurable sets. (Contributed by Thierry Arnoux, 21-Jan-2017.)
Hypotheses
Ref Expression
orvccel.1  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
orvccel.2  |-  ( ph  ->  J  e.  Top )
orvccel.3  |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
) )
orvccel.4  |-  ( ph  ->  A  e.  V )
orvcoel.5  |-  ( ph  ->  { y  e.  U. J  |  y R A }  e.  J
)
Assertion
Ref Expression
orvcoel  |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S
)
Distinct variable groups:    y, A    y, R    y, X    y, J
Allowed substitution hints:    ph( y)    S( y)    V( y)

Proof of Theorem orvcoel
StepHypRef Expression
1 orvccel.1 . . 3  |-  ( ph  ->  S  e.  U. ran sigAlgebra )
2 orvccel.2 . . 3  |-  ( ph  ->  J  e.  Top )
3 orvccel.3 . . 3  |-  ( ph  ->  X  e.  ( SMblFnM (sigaGen `  J )
) )
4 orvccel.4 . . 3  |-  ( ph  ->  A  e.  V )
51, 2, 3, 4orvcval4 29302 . 2  |-  ( ph  ->  ( XRV/𝑐 R A )  =  ( `' X " { y  e.  U. J  | 
y R A }
) )
62sgsiga 28973 . . 3  |-  ( ph  ->  (sigaGen `  J )  e.  U. ran sigAlgebra )
7 sssigagen 28976 . . . . 5  |-  ( J  e.  Top  ->  J  C_  (sigaGen `  J )
)
82, 7syl 17 . . . 4  |-  ( ph  ->  J  C_  (sigaGen `  J
) )
9 orvcoel.5 . . . 4  |-  ( ph  ->  { y  e.  U. J  |  y R A }  e.  J
)
108, 9sseldd 3465 . . 3  |-  ( ph  ->  { y  e.  U. J  |  y R A }  e.  (sigaGen `  J ) )
111, 6, 3, 10mbfmcnvima 29088 . 2  |-  ( ph  ->  ( `' X " { y  e.  U. J  |  y R A } )  e.  S
)
125, 11eqeltrd 2507 1  |-  ( ph  ->  ( XRV/𝑐 R A )  e.  S
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1872   {crab 2775    C_ wss 3436   U.cuni 4219   class class class wbr 4423   `'ccnv 4852   ran crn 4854   "cima 4856   ` cfv 5601  (class class class)co 6306   Topctop 19916  sigAlgebracsiga 28938  sigaGencsigagen 28969  MblFnMcmbfm 29081  ∘RV/𝑐corvc 29297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-1st 6808  df-2nd 6809  df-map 7486  df-siga 28939  df-sigagen 28970  df-mbfm 29082  df-orvc 29298
This theorem is referenced by:  orrvcoel  29307
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