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Theorem salgenuni 45053
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 45037 . . . . 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 2773 . . 3 (πœ‘ β†’ 𝑆 = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
76unieqd 4923 . 2 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
8 ssrab2 4078 . . . 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 45047 . . . 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 4920 . . . . . . . . . 10 (𝑠 = 𝑑 β†’ βˆͺ 𝑠 = βˆͺ 𝑑)
1312eqeq1d 2735 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑑 = βˆͺ 𝑋))
14 sseq2 4009 . . . . . . . . 9 (𝑠 = 𝑑 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑑))
1513, 14anbi12d 632 . . . . . . . 8 (𝑠 = 𝑑 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝑑 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑑)))
1615elrab 3684 . . . . . . 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 771 . . . . 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 2742 . . . . . 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 2773 . . . 4 (𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑑 = π‘ˆ)
2322adantl 483 . . 3 ((πœ‘ ∧ 𝑑 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ βˆͺ 𝑑 = π‘ˆ)
249, 11, 23intsaluni 45045 . 2 (πœ‘ β†’ βˆͺ ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} = π‘ˆ)
257, 24eqtrd 2773 1 (πœ‘ β†’ βˆͺ 𝑆 = π‘ˆ)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909  βˆ© cint 4951  β€˜cfv 6544  SAlgcsalg 45024  SalGencsalgen 45028
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-salg 45025  df-salgen 45029
This theorem is referenced by:  unisalgen  45056  dfsalgen2  45057
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