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Theorem salgenuni 41483
 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 41469 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2814 . . 3 (𝜑𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
76unieqd 4681 . 2 (𝜑 𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
8 ssrab2 3908 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
98a1i 11 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
10 salgenn0 41477 . . . 4 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
113, 10syl 17 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
12 unieq 4679 . . . . . . . . . 10 (𝑠 = 𝑡 𝑠 = 𝑡)
1312eqeq1d 2780 . . . . . . . . 9 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
14 sseq2 3846 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
1513, 14anbi12d 624 . . . . . . . 8 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
1615elrab 3572 . . . . . . 7 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1716biimpi 208 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1817simprld 762 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
19 salgenuni.u . . . . . . 7 𝑈 = 𝑋
2019eqcomi 2787 . . . . . 6 𝑋 = 𝑈
2120a1i 11 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋 = 𝑈)
2218, 21eqtrd 2814 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑈)
2322adantl 475 . . 3 ((𝜑𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑈)
249, 11, 23intsaluni 41475 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑈)
257, 24eqtrd 2814 1 (𝜑 𝑆 = 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   ≠ wne 2969  {crab 3094   ⊆ wss 3792  ∅c0 4141  ∪ cuni 4671  ∩ cint 4710  ‘cfv 6135  SAlgcsalg 41456  SalGencsalgen 41460 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-salg 41457  df-salgen 41461 This theorem is referenced by:  unisalgen  41486  dfsalgen2  41487
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