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Theorem unisalgen2 46369
Description: The union of a set belongs is equal to the union of the sigma-algebra generated by the set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
unisalgen2.x (𝜑𝐴𝑉)
unisalgen2.s 𝑆 = (SalGen‘𝐴)
Assertion
Ref Expression
unisalgen2 (𝜑 𝑆 = 𝐴)

Proof of Theorem unisalgen2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 unisalgen2.s . . . . . 6 𝑆 = (SalGen‘𝐴)
21eqcomi 2746 . . . . 5 (SalGen‘𝐴) = 𝑆
32a1i 11 . . . 4 (𝜑 → (SalGen‘𝐴) = 𝑆)
4 unisalgen2.x . . . . 5 (𝜑𝐴𝑉)
54dfsalgen2 46356 . . . 4 (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆) ∧ ∀𝑥 ∈ SAlg (( 𝑥 = 𝐴𝐴𝑥) → 𝑆𝑥))))
63, 5mpbid 232 . . 3 (𝜑 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆) ∧ ∀𝑥 ∈ SAlg (( 𝑥 = 𝐴𝐴𝑥) → 𝑆𝑥)))
76simpld 494 . 2 (𝜑 → (𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆))
87simp2d 1144 1 (𝜑 𝑆 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951   cuni 4907  cfv 6561  SAlgcsalg 46323  SalGencsalgen 46327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-salg 46324  df-salgen 46328
This theorem is referenced by:  incsmf  46757  decsmf  46782
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