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Theorem sssalgen 46355
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 4963 . . . 4 (𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑋𝑡)
2 unieq 4917 . . . . . . . . 9 (𝑠 = 𝑡 𝑠 = 𝑡)
32eqeq1d 2738 . . . . . . . 8 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
4 sseq2 4009 . . . . . . . 8 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
53, 4anbi12d 632 . . . . . . 7 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
65elrab 3691 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
76biimpi 216 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
87simprrd 773 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑡)
91, 8mprgbir 3067 . . 3 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}
109a1i 11 . 2 (𝑋𝑉𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
11 sssalgen.1 . . 3 𝑆 = (SalGen‘𝑋)
12 salgenval 46341 . . 3 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
1311, 12eqtr2id 2789 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑆)
1410, 13sseqtrd 4019 1 (𝑋𝑉𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435  wss 3950   cuni 4906   cint 4945  cfv 6560  SAlgcsalg 46328  SalGencsalgen 46332
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-salg 46329  df-salgen 46333
This theorem is referenced by:  dfsalgen2  46361  iooborel  46371  opnssborel  46655  cnfsmf  46760
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