Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isrnsigaOLD Structured version   Visualization version   GIF version

Theorem isrnsigaOLD 30303
 Description: The property of being a sigma-algebra on an indefinite base set. (Contributed by Thierry Arnoux, 3-Sep-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
isrnsigaOLD (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
Distinct variable group:   𝑥,𝑜,𝑆

Proof of Theorem isrnsigaOLD
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 df-siga 30299 . . 3 sigAlgebra = (𝑜 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))})
2 df-rab 2950 . . . . 5 {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} = {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
3 vex 3234 . . . . . . . 8 𝑠 ∈ V
4 elpwg 4199 . . . . . . . 8 (𝑠 ∈ V → (𝑠 ∈ 𝒫 𝒫 𝑜𝑠 ⊆ 𝒫 𝑜))
53, 4ax-mp 5 . . . . . . 7 (𝑠 ∈ 𝒫 𝒫 𝑜𝑠 ⊆ 𝒫 𝑜)
65anbi1i 731 . . . . . 6 ((𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))) ↔ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))))
76abbii 2768 . . . . 5 {𝑠 ∣ (𝑠 ∈ 𝒫 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} = {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))}
82, 7eqtr2i 2674 . . . 4 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} = {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))}
9 grothpwex 9687 . . . . . 6 𝒫 𝑜 ∈ V
109pwex 4878 . . . . 5 𝒫 𝒫 𝑜 ∈ V
1110rabex 4845 . . . 4 {𝑠 ∈ 𝒫 𝒫 𝑜 ∣ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))} ∈ V
128, 11eqeltri 2726 . . 3 {𝑠 ∣ (𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)))} ∈ V
13 sseq1 3659 . . . 4 (𝑠 = 𝑆 → (𝑠 ⊆ 𝒫 𝑜𝑆 ⊆ 𝒫 𝑜))
14 eleq2 2719 . . . . 5 (𝑠 = 𝑆 → (𝑜𝑠𝑜𝑆))
15 eleq2 2719 . . . . . 6 (𝑠 = 𝑆 → ((𝑜𝑥) ∈ 𝑠 ↔ (𝑜𝑥) ∈ 𝑆))
1615raleqbi1dv 3176 . . . . 5 (𝑠 = 𝑆 → (∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ↔ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
17 pweq 4194 . . . . . 6 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
18 biidd 252 . . . . . . 7 (𝑠 = 𝑆 → (𝑥 ≼ ω ↔ 𝑥 ≼ ω))
19 eleq2 2719 . . . . . . 7 (𝑠 = 𝑆 → ( 𝑥𝑠 𝑥𝑆))
2018, 19imbi12d 333 . . . . . 6 (𝑠 = 𝑆 → ((𝑥 ≼ ω → 𝑥𝑠) ↔ (𝑥 ≼ ω → 𝑥𝑆)))
2117, 20raleqbidv 3182 . . . . 5 (𝑠 = 𝑆 → (∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠) ↔ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
2214, 16, 213anbi123d 1439 . . . 4 (𝑠 = 𝑆 → ((𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠)) ↔ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
2313, 22anbi12d 747 . . 3 (𝑠 = 𝑆 → ((𝑠 ⊆ 𝒫 𝑜 ∧ (𝑜𝑠 ∧ ∀𝑥𝑠 (𝑜𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠(𝑥 ≼ ω → 𝑥𝑠))) ↔ (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
241, 12, 23abfmpunirn 29580 . 2 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
25 rexv 3251 . . 3 (∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) ↔ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
2625anbi2i 730 . 2 ((𝑆 ∈ V ∧ ∃𝑜 ∈ V (𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))) ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
2724, 26bitri 264 1 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  {cab 2637  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468   class class class wbr 4685  ran crn 5144  ωcom 7107   ≼ cdom 7995  sigAlgebracsiga 30298 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-groth 9683 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-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-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-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-siga 30299 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator