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Theorem salgenuni 39862
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 39848 . . . . 5 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
53, 4syl 17 . . . 4 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
62, 5eqtrd 2655 . . 3 (𝜑𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
76unieqd 4412 . 2 (𝜑 𝑆 = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
8 ssrab2 3666 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
98a1i 11 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
10 salgenn0 39856 . . . 4 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
113, 10syl 17 . . 3 (𝜑 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
12 unieq 4410 . . . . . . . . . 10 (𝑠 = 𝑡 𝑠 = 𝑡)
1312eqeq1d 2623 . . . . . . . . 9 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
14 sseq2 3606 . . . . . . . . 9 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
1513, 14anbi12d 746 . . . . . . . 8 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
1615elrab 3346 . . . . . . 7 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1716biimpi 206 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1817simprld 794 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
19 salgenuni.u . . . . . . 7 𝑈 = 𝑋
2019eqcomi 2630 . . . . . 6 𝑋 = 𝑈
2120a1i 11 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋 = 𝑈)
2218, 21eqtrd 2655 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑈)
2322adantl 482 . . 3 ((𝜑𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑈)
249, 11, 23intsaluni 39854 . 2 (𝜑 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} = 𝑈)
257, 24eqtrd 2655 1 (𝜑 𝑆 = 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wne 2790  {crab 2911  wss 3555  c0 3891   cuni 4402   cint 4440  cfv 5847  SAlgcsalg 39835  SalGencsalgen 39839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-salg 39836  df-salgen 39840
This theorem is referenced by:  unisalgen  39865  dfsalgen2  39866
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