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Theorem sgsiga 30545
Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Hypothesis
Ref Expression
sgsiga.1 (𝜑𝐴𝑉)
Assertion
Ref Expression
sgsiga (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)

Proof of Theorem sgsiga
StepHypRef Expression
1 sgsiga.1 . 2 (𝜑𝐴𝑉)
2 sigagensiga 30544 . 2 (𝐴𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴))
3 elrnsiga 30529 . 2 ((sigaGen‘𝐴) ∈ (sigAlgebra‘ 𝐴) → (sigaGen‘𝐴) ∈ ran sigAlgebra)
41, 2, 33syl 18 1 (𝜑 → (sigaGen‘𝐴) ∈ ran sigAlgebra)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145   cuni 4574  ran crn 5250  cfv 6031  sigAlgebracsiga 30510  sigaGencsigagen 30541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-siga 30511  df-sigagen 30542
This theorem is referenced by:  elsigagen2  30551  cldssbrsiga  30590  mbfmbfm  30660  imambfm  30664  sxbrsigalem2  30688  sxbrsiga  30692  sibf0  30736  sibff  30738  sibfinima  30741  sibfof  30742  sitgclg  30744  orvcval4  30862  orvcoel  30863  orvccel  30864
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