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Theorem unisalgen2 46309
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 2743 . . . . 5 (SalGen‘𝐴) = 𝑆
32a1i 11 . . . 4 (𝜑 → (SalGen‘𝐴) = 𝑆)
4 unisalgen2.x . . . . 5 (𝜑𝐴𝑉)
54dfsalgen2 46296 . . . 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 1142 1 (𝜑 𝑆 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wss 3962   cuni 4911  cfv 6562  SAlgcsalg 46263  SalGencsalgen 46267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-salg 46264  df-salgen 46268
This theorem is referenced by:  incsmf  46697  decsmf  46722
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