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Theorem unisalgen2 40890
 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 2660 . . . . 5 (SalGen‘𝐴) = 𝑆
32a1i 11 . . . 4 (𝜑 → (SalGen‘𝐴) = 𝑆)
4 unisalgen2.x . . . . 5 (𝜑𝐴𝑉)
54dfsalgen2 40877 . . . 4 (𝜑 → ((SalGen‘𝐴) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆) ∧ ∀𝑥 ∈ SAlg (( 𝑥 = 𝐴𝐴𝑥) → 𝑆𝑥))))
63, 5mpbid 222 . . 3 (𝜑 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆) ∧ ∀𝑥 ∈ SAlg (( 𝑥 = 𝐴𝐴𝑥) → 𝑆𝑥)))
76simpld 474 . 2 (𝜑 → (𝑆 ∈ SAlg ∧ 𝑆 = 𝐴𝐴𝑆))
87simp2d 1094 1 (𝜑 𝑆 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ⊆ wss 3607  ∪ cuni 4468  ‘cfv 5926  SAlgcsalg 40846  SalGencsalgen 40850 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-salg 40847  df-salgen 40851 This theorem is referenced by:  incsmf  41272  decsmf  41296
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