Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  issalgend Structured version   Visualization version   GIF version

Theorem issalgend 39232
 Description: One side of dfsalgen2 39235. If a sigma-algebra on ∪ 𝑋 includes 𝑋 and it is included in all the sigma-algebras with such two properties, then it is the sigma-algebra generated by 𝑋. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypotheses
Ref Expression
issalgend.x (𝜑𝑋𝑉)
issalgend.s (𝜑𝑆 ∈ SAlg)
issalgend.u (𝜑 𝑆 = 𝑋)
issalgend.i (𝜑𝑋𝑆)
issalgend.a ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
Assertion
Ref Expression
issalgend (𝜑 → (SalGen‘𝑋) = 𝑆)
Distinct variable groups:   𝑦,𝑆   𝑦,𝑋   𝜑,𝑦
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem issalgend
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 issalgend.x . . 3 (𝜑𝑋𝑉)
2 eqid 2610 . . 3 (SalGen‘𝑋) = (SalGen‘𝑋)
3 issalgend.s . . 3 (𝜑𝑆 ∈ SAlg)
4 issalgend.i . . 3 (𝜑𝑋𝑆)
5 issalgend.u . . 3 (𝜑 𝑆 = 𝑋)
61, 2, 3, 4, 5salgenss 39230 . 2 (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7 simpl 472 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝜑)
8 elrabi 3328 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 ∈ SAlg)
98adantl 481 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 ∈ SAlg)
10 unieq 4380 . . . . . . . . . . . 12 (𝑠 = 𝑦 𝑠 = 𝑦)
1110eqeq1d 2612 . . . . . . . . . . 11 (𝑠 = 𝑦 → ( 𝑠 = 𝑋 𝑦 = 𝑋))
12 sseq2 3590 . . . . . . . . . . 11 (𝑠 = 𝑦 → (𝑋𝑠𝑋𝑦))
1311, 12anbi12d 743 . . . . . . . . . 10 (𝑠 = 𝑦 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑦 = 𝑋𝑋𝑦)))
1413elrab 3331 . . . . . . . . 9 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1514biimpi 205 . . . . . . . 8 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1615simprld 791 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 = 𝑋)
1716adantl 481 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 = 𝑋)
1815simprrd 793 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑦)
1918adantl 481 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑋𝑦)
20 issalgend.a . . . . . 6 ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
217, 9, 17, 19, 20syl13anc 1320 . . . . 5 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑆𝑦)
2221ralrimiva 2949 . . . 4 (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
23 ssint 4428 . . . 4 (𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
2422, 23sylibr 223 . . 3 (𝜑𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
25 salgenval 39217 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
261, 25syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2724, 26sseqtr4d 3605 . 2 (𝜑𝑆 ⊆ (SalGen‘𝑋))
286, 27eqssd 3585 1 (𝜑 → (SalGen‘𝑋) = 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  ∪ cuni 4372  ∩ cint 4410  ‘cfv 5804  SAlgcsalg 39204  SalGencsalgen 39208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-salg 39205  df-salgen 39209 This theorem is referenced by:  dfsalgen2  39235
 Copyright terms: Public domain W3C validator