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Theorem salgencl 43413
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 43404 . 2 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2 ssrab2 3969 . . . 4 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg
32a1i 11 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ⊆ SAlg)
4 salgenn0 43412 . . 3 (𝑋𝑉 → {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ≠ ∅)
5 unieq 4807 . . . . . . . . 9 (𝑠 = 𝑡 𝑠 = 𝑡)
65eqeq1d 2740 . . . . . . . 8 (𝑠 = 𝑡 → ( 𝑠 = 𝑋 𝑡 = 𝑋))
7 sseq2 3903 . . . . . . . 8 (𝑠 = 𝑡 → (𝑋𝑠𝑋𝑡))
86, 7anbi12d 634 . . . . . . 7 (𝑠 = 𝑡 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑡 = 𝑋𝑋𝑡)))
98elrab 3588 . . . . . 6 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
109biimpi 219 . . . . 5 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑡 ∈ SAlg ∧ ( 𝑡 = 𝑋𝑋𝑡)))
1110simprld 772 . . . 4 (𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑡 = 𝑋)
1211adantl 485 . . 3 ((𝑋𝑉𝑡 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑡 = 𝑋)
133, 4, 12intsal 43411 . 2 (𝑋𝑉 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ∈ SAlg)
141, 13eqeltrd 2833 1 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {crab 3057  wss 3843   cuni 4796   cint 4836  cfv 6339  SAlgcsalg 43391  SalGencsalgen 43395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-int 4837  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-salg 43392  df-salgen 43396
This theorem is referenced by:  unisalgen  43421  dfsalgen2  43422  salgencld  43430
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