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Theorem salgencl 39873
Description: SalGen actually generates a sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Assertion
Ref Expression
salgencl (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)

Proof of Theorem salgencl
Dummy variables 𝑡 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 salgenval 39864 . 2 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2 ssrab2 3668 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
32a1i 11 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
4 salgenn0 39872 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
5 unieq 4412 . . . . . . . . 9 (𝑠 = 𝑡 𝑠 = 𝑡)
65eqeq1d 2623 . . . . . . . 8 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
7 sseq2 3608 . . . . . . . 8 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
86, 7anbi12d 746 . . . . . . 7 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
98elrab 3347 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
109biimpi 206 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1110simprld 794 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
1211adantl 482 . . 3 ((𝑋𝑉𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑋)
133, 4, 12intsal 39871 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ SAlg)
141, 13eqeltrd 2698 1 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2911  wss 3556   cuni 4404   cint 4442  cfv 5849  SAlgcsalg 39851  SalGencsalgen 39855
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 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
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 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-salg 39852  df-salgen 39856
This theorem is referenced by:  unisalgen  39881  dfsalgen2  39882  salgencld  39890
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