Theorem sgsiga 34176

Description: A generated sigma-algebra is a sigma-algebra. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypothesis

Ref	Expression
sgsiga.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
sgsiga	⊢ (𝜑 → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)

Proof of Theorem sgsiga

Step	Hyp	Ref	Expression
1		sgsiga.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		sigagensiga 34175	. 2 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴))
3		elrnsiga 34160	. 2 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
4	1, 2, 3	3syl 18	1 ⊢ (𝜑 → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)

Syntax hints:   → wi 4   ∈ wcel 2113  ∪ cuni 4858  ran crn 5620  ‘cfv 6486  sigAlgebracsiga 34142  sigaGencsigagen 34172

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fv 6494  df-siga 34143  df-sigagen 34173

This theorem is referenced by:  elsigagen2  34182  cldssbrsiga  34221  imambfm  34296  sxbrsigalem2  34320  sxbrsiga  34324  sibf0  34368  sibff  34370  sibfinima  34373  sibfof  34374  sitgclg  34376  orvcval4  34495  orvcoel  34496  orvccel  34497