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Theorem sssalgen 46291
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 4969 . . . 4 (𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑋𝑡)
2 unieq 4923 . . . . . . . . 9 (𝑠 = 𝑡 𝑠 = 𝑡)
32eqeq1d 2737 . . . . . . . 8 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
4 sseq2 4022 . . . . . . . 8 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
53, 4anbi12d 632 . . . . . . 7 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
65elrab 3695 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
76biimpi 216 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
87simprrd 774 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑡)
91, 8mprgbir 3066 . . 3 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}
109a1i 11 . 2 (𝑋𝑉𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
11 sssalgen.1 . . 3 𝑆 = (SalGen‘𝑋)
12 salgenval 46277 . . 3 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
1311, 12eqtr2id 2788 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑆)
1410, 13sseqtrd 4036 1 (𝑋𝑉𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  wss 3963   cuni 4912   cint 4951  cfv 6563  SAlgcsalg 46264  SalGencsalgen 46268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-salg 46265  df-salgen 46269
This theorem is referenced by:  dfsalgen2  46297  iooborel  46307  opnssborel  46591  cnfsmf  46696
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