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Theorem sssalgen 42970
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 4854 . . . 4 (𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑋𝑡)
2 unieq 4811 . . . . . . . . 9 (𝑠 = 𝑡 𝑠 = 𝑡)
32eqeq1d 2800 . . . . . . . 8 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
4 sseq2 3941 . . . . . . . 8 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
53, 4anbi12d 633 . . . . . . 7 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
65elrab 3628 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
76biimpi 219 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
87simprrd 773 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑡)
91, 8mprgbir 3121 . . 3 𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}
109a1i 11 . 2 (𝑋𝑉𝑋 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
11 sssalgen.1 . . 3 𝑆 = (SalGen‘𝑋)
12 salgenval 42958 . . 3 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
1311, 12syl5req 2846 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑆)
1410, 13sseqtrd 3955 1 (𝑋𝑉𝑋𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  {crab 3110  wss 3881   cuni 4800   cint 4838  cfv 6324  SAlgcsalg 42945  SalGencsalgen 42949
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-salg 42946  df-salgen 42950
This theorem is referenced by:  dfsalgen2  42976  iooborel  42986  opnssborel  43269  cnfsmf  43369
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