Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigagenss2 Structured version   Visualization version   GIF version

Theorem sigagenss2 30187
Description: Sufficient condition for inclusion of sigma-algebras. This is used to prove equality of sigma-algebras. (Contributed by Thierry Arnoux, 10-Oct-2017.)
Assertion
Ref Expression
sigagenss2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))

Proof of Theorem sigagenss2
StepHypRef Expression
1 sigagensiga 30178 . . . 4 (𝐵𝑉 → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐵))
213ad2ant3 1082 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐵))
3 simp1 1059 . . . 4 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 = 𝐵)
43fveq2d 6182 . . 3 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigAlgebra‘ 𝐴) = (sigAlgebra‘ 𝐵))
52, 4eleqtrrd 2702 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴))
6 simp2 1060 . 2 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → 𝐴 ⊆ (sigaGen‘𝐵))
7 sigagenss 30186 . 2 (((sigaGen‘𝐵) ∈ (sigAlgebra‘ 𝐴) ∧ 𝐴 ⊆ (sigaGen‘𝐵)) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
85, 6, 7syl2anc 692 1 (( 𝐴 = 𝐵𝐴 ⊆ (sigaGen‘𝐵) ∧ 𝐵𝑉) → (sigaGen‘𝐴) ⊆ (sigaGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1481  wcel 1988  wss 3567   cuni 4427  cfv 5876  sigAlgebracsiga 30144  sigaGencsigagen 30175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-siga 30145  df-sigagen 30176
This theorem is referenced by:  sxbrsigalem3  30308  sxbrsigalem1  30321  sxbrsigalem2  30322  sxbrsigalem4  30323  sxbrsigalem5  30324  sxbrsiga  30326
  Copyright terms: Public domain W3C validator