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Theorem unisalgen2 44681
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 (πœ‘ β†’ 𝐴 ∈ 𝑉)
unisalgen2.s 𝑆 = (SalGenβ€˜π΄)
Assertion
Ref Expression
unisalgen2 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ 𝐴)

Proof of Theorem unisalgen2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 unisalgen2.s . . . . . 6 𝑆 = (SalGenβ€˜π΄)
21eqcomi 2742 . . . . 5 (SalGenβ€˜π΄) = 𝑆
32a1i 11 . . . 4 (πœ‘ β†’ (SalGenβ€˜π΄) = 𝑆)
4 unisalgen2.x . . . . 5 (πœ‘ β†’ 𝐴 ∈ 𝑉)
54dfsalgen2 44668 . . . 4 (πœ‘ β†’ ((SalGenβ€˜π΄) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝐴 ∧ 𝐴 βŠ† 𝑆) ∧ βˆ€π‘₯ ∈ SAlg ((βˆͺ π‘₯ = βˆͺ 𝐴 ∧ 𝐴 βŠ† π‘₯) β†’ 𝑆 βŠ† π‘₯))))
63, 5mpbid 231 . . 3 (πœ‘ β†’ ((𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝐴 ∧ 𝐴 βŠ† 𝑆) ∧ βˆ€π‘₯ ∈ SAlg ((βˆͺ π‘₯ = βˆͺ 𝐴 ∧ 𝐴 βŠ† π‘₯) β†’ 𝑆 βŠ† π‘₯)))
76simpld 496 . 2 (πœ‘ β†’ (𝑆 ∈ SAlg ∧ βˆͺ 𝑆 = βˆͺ 𝐴 ∧ 𝐴 βŠ† 𝑆))
87simp2d 1144 1 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  βˆͺ cuni 4866  β€˜cfv 6497  SAlgcsalg 44635  SalGencsalgen 44639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-salg 44636  df-salgen 44640
This theorem is referenced by:  incsmf  45069  decsmf  45094
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