Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sssalgen Structured version   Visualization version   GIF version

Theorem sssalgen 44546
Description: A set is a subset of the sigma-algebra it generates. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypothesis
Ref Expression
sssalgen.1 𝑆 = (SalGen‘𝑋)
Assertion
Ref Expression
sssalgen (𝑋𝑉𝑋𝑆)

Proof of Theorem sssalgen
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssint 4924 . . . 4 (𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑋𝑡)
2 unieq 4875 . . . . . . . . 9 (𝑠 = 𝑡 𝑠 = 𝑡)
32eqeq1d 2738 . . . . . . . 8 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
4 sseq2 3969 . . . . . . . 8 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
53, 4anbi12d 631 . . . . . . 7 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
65elrab 3644 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
76biimpi 215 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
87simprrd 772 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑡)
91, 8mprgbir 3070 . . 3 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}
109a1i 11 . 2 (𝑋𝑉𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
11 sssalgen.1 . . 3 𝑆 = (SalGen‘𝑋)
12 salgenval 44532 . . 3 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
1311, 12eqtr2id 2789 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑆)
1410, 13sseqtrd 3983 1 (𝑋𝑉𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3406  wss 3909   cuni 4864   cint 4906  cfv 6494  SAlgcsalg 44519  SalGencsalgen 44523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7669
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6446  df-fun 6496  df-fv 6502  df-salg 44520  df-salgen 44524
This theorem is referenced by:  dfsalgen2  44552  iooborel  44562  opnssborel  44846  cnfsmf  44951
  Copyright terms: Public domain W3C validator