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Theorem sgsiga 27770
Description: A generated sigma algebra is a sigma algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypothesis
Ref Expression
sgsiga.1  |-  ( ph  ->  A  e.  V )
Assertion
Ref Expression
sgsiga  |-  ( ph  ->  (sigaGen `  A )  e.  U. ran sigAlgebra )

Proof of Theorem sgsiga
StepHypRef Expression
1 sgsiga.1 . 2  |-  ( ph  ->  A  e.  V )
2 sigagensiga 27769 . 2  |-  ( A  e.  V  ->  (sigaGen `  A )  e.  (sigAlgebra ` 
U. A ) )
3 elrnsiga 27754 . 2  |-  ( (sigaGen `  A )  e.  (sigAlgebra ` 
U. A )  -> 
(sigaGen `  A )  e. 
U. ran sigAlgebra )
41, 2, 33syl 20 1  |-  ( ph  ->  (sigaGen `  A )  e.  U. ran sigAlgebra )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1762   U.cuni 4240   ran crn 4995   ` cfv 5581  sigAlgebracsiga 27735  sigaGencsigagen 27766
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-int 4278  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fv 5589  df-siga 27736  df-sigagen 27767
This theorem is referenced by:  elsigagen2  27776  cldssbrsiga  27786  mbfmbfm  27857  imambfm  27861  sxbrsigalem2  27885  sxbrsiga  27889  sibf0  27904  sibff  27906  sibfinima  27909  sibfof  27910  sitgclg  27912  orvcval4  28027  orvcoel  28028  orvccel  28029
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