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Theorem unisalgen2 41303
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 2806 . . . . 5 (SalGen‘𝐴) = 𝑆
32a1i 11 . . . 4 (𝜑 → (SalGen‘𝐴) = 𝑆)
4 unisalgen2.x . . . . 5 (𝜑𝐴𝑉)
54dfsalgen2 41290 . . . 4 (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆) ∧ ∀𝑥 ∈ SAlg (( 𝑥 = 𝐴𝐴𝑥) → 𝑆𝑥))))
63, 5mpbid 224 . . 3 (𝜑 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆) ∧ ∀𝑥 ∈ SAlg (( 𝑥 = 𝐴𝐴𝑥) → 𝑆𝑥)))
76simpld 489 . 2 (𝜑 → (𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆))
87simp2d 1174 1 (𝜑 𝑆 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3087  wss 3767   cuni 4626  cfv 6099  SAlgcsalg 41259  SalGencsalgen 41263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-int 4666  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-salg 41260  df-salgen 41264
This theorem is referenced by:  incsmf  41685  decsmf  41709
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