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Theorem salgenuni 43876
Description: The base set of the sigma-algebra generated by a set is the union of the set itself. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
salgenuni.x (𝜑𝑋𝑉)
salgenuni.s 𝑆 = (SalGen‘𝑋)
salgenuni.u 𝑈 = 𝑋
Assertion
Ref Expression
salgenuni (𝜑 𝑆 = 𝑈)

Proof of Theorem salgenuni
Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 salgenuni.s . . . . 5 𝑆 = (SalGen‘𝑋)
21a1i 11 . . . 4 (𝜑𝑆 = (SalGen‘𝑋))
3 salgenuni.x . . . . 5 (𝜑𝑋𝑉)
4 salgenval 43862 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2778 . . 3 (𝜑𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
76unieqd 4853 . 2 (𝜑 𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
8 ssrab2 4013 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
98a1i 11 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
10 salgenn0 43870 . . . 4 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
113, 10syl 17 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
12 unieq 4850 . . . . . . . . . 10 (𝑠 = 𝑡 𝑠 = 𝑡)
1312eqeq1d 2740 . . . . . . . . 9 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
14 sseq2 3947 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
1513, 14anbi12d 631 . . . . . . . 8 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
1615elrab 3624 . . . . . . 7 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1716biimpi 215 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1817simprld 769 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
19 salgenuni.u . . . . . . 7 𝑈 = 𝑋
2019eqcomi 2747 . . . . . 6 𝑋 = 𝑈
2120a1i 11 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋 = 𝑈)
2218, 21eqtrd 2778 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑈)
2322adantl 482 . . 3 ((𝜑𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑈)
249, 11, 23intsaluni 43868 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑈)
257, 24eqtrd 2778 1 (𝜑 𝑆 = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  {crab 3068  wss 3887  c0 4256   cuni 4839   cint 4879  cfv 6433  SAlgcsalg 43849  SalGencsalgen 43853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-salg 43850  df-salgen 43854
This theorem is referenced by:  unisalgen  43879  dfsalgen2  43880
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