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Theorem salgenuni 45351
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 45335 . . . . 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 2770 . . 3 (πœ‘ β†’ 𝑆 = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
76unieqd 4921 . 2 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
8 ssrab2 4076 . . . 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 45345 . . . 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 4918 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ βˆͺ 𝑠 = βˆͺ 𝑑)
1312eqeq1d 2732 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑑 = βˆͺ 𝑋))
14 sseq2 4007 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑑))
1513, 14anbi12d 629 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝑑 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑑)))
1615elrab 3682 . . . . . . 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 768 . . . . 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 2739 . . . . . 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 2770 . . . 4 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑑 = π‘ˆ)
2322adantl 480 . . 3 ((πœ‘ ∧ 𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ βˆͺ 𝑑 = π‘ˆ)
249, 11, 23intsaluni 45343 . 2 (πœ‘ β†’ βˆͺ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} = π‘ˆ)
257, 24eqtrd 2770 1 (πœ‘ β†’ βˆͺ 𝑆 = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  {crab 3430   βŠ† wss 3947  βˆ…c0 4321  βˆͺ cuni 4907  βˆ© cint 4949  β€˜cfv 6542  SAlgcsalg 45322  SalGencsalgen 45326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-salg 45323  df-salgen 45327
This theorem is referenced by:  unisalgen  45354  dfsalgen2  45355
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