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Theorem salgenuni 44731
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 (πœ‘ β†’ 𝑋 ∈ 𝑉)
salgenuni.s 𝑆 = (SalGenβ€˜π‘‹)
salgenuni.u π‘ˆ = βˆͺ 𝑋
Assertion
Ref Expression
salgenuni (πœ‘ β†’ βˆͺ 𝑆 = π‘ˆ)
Proof of Theorem salgenuni
Dummy variables 𝑑 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 salgenuni.s . . . . 5 𝑆 = (SalGenβ€˜π‘‹)
21a1i 11 . . . 4 (πœ‘ β†’ 𝑆 = (SalGenβ€˜π‘‹))
3 salgenuni.x . . . . 5 (πœ‘ β†’ 𝑋 ∈ 𝑉)
4 salgenval 44715 . . . . 5 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
53, 4syl 17 . . . 4 (πœ‘ β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
62, 5eqtrd 2771 . . 3 (πœ‘ β†’ 𝑆 = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
76unieqd 4899 . 2 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
8 ssrab2 4057 . . . 4 {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} βŠ† SAlg
98a1i 11 . . 3 (πœ‘ β†’ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} βŠ† SAlg)
10 salgenn0 44725 . . . 4 (𝑋 ∈ 𝑉 β†’ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β‰  βˆ…)
113, 10syl 17 . . 3 (πœ‘ β†’ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β‰  βˆ…)
12 unieq 4896 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ βˆͺ 𝑠 = βˆͺ 𝑑)
1312eqeq1d 2733 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑑 = βˆͺ 𝑋))
14 sseq2 3988 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑑))
1513, 14anbi12d 631 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝑑 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑑)))
1615elrab 3663 . . . . . . 7 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ↔ (𝑑 ∈ SAlg ∧ (βˆͺ 𝑑 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑑)))
1716biimpi 215 . . . . . 6 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ (𝑑 ∈ SAlg ∧ (βˆͺ 𝑑 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑑)))
1817simprld 770 . . . . 5 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑑 = βˆͺ 𝑋)
19 salgenuni.u . . . . . . 7 π‘ˆ = βˆͺ 𝑋
2019eqcomi 2740 . . . . . 6 βˆͺ 𝑋 = π‘ˆ
2120a1i 11 . . . . 5 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑋 = π‘ˆ)
2218, 21eqtrd 2771 . . . 4 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑑 = π‘ˆ)
2322adantl 482 . . 3 ((πœ‘ ∧ 𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ βˆͺ 𝑑 = π‘ˆ)
249, 11, 23intsaluni 44723 . 2 (πœ‘ β†’ βˆͺ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} = π‘ˆ)
257, 24eqtrd 2771 1 (πœ‘ β†’ βˆͺ 𝑆 = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2939  {crab 3418   βŠ† wss 3928  βˆ…c0 4302  βˆͺ cuni 4885  βˆ© cint 4927  β€˜cfv 6516  SAlgcsalg 44702  SalGencsalgen 44706
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 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
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 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-int 4928  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-iota 6468  df-fun 6518  df-fv 6524  df-salg 44703  df-salgen 44707
This theorem is referenced by:  unisalgen  44734  dfsalgen2  44735
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