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

Theorem issalgend 39879
Description: One side of dfsalgen2 39882. 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 2621 . . 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 39877 . 2 (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7 simpl 473 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝜑)
8 elrabi 3343 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 ∈ SAlg)
98adantl 482 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 ∈ SAlg)
10 unieq 4412 . . . . . . . . . . . 12 (𝑠 = 𝑦 𝑠 = 𝑦)
1110eqeq1d 2623 . . . . . . . . . . 11 (𝑠 = 𝑦 → ( 𝑠 = 𝑋 𝑦 = 𝑋))
12 sseq2 3608 . . . . . . . . . . 11 (𝑠 = 𝑦 → (𝑋𝑠𝑋𝑦))
1311, 12anbi12d 746 . . . . . . . . . 10 (𝑠 = 𝑦 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑦 = 𝑋𝑋𝑦)))
1413elrab 3347 . . . . . . . . 9 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1514biimpi 206 . . . . . . . 8 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1615simprld 794 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 = 𝑋)
1716adantl 482 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 = 𝑋)
1815simprrd 796 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑦)
1918adantl 482 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑋𝑦)
20 issalgend.a . . . . . 6 ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
217, 9, 17, 19, 20syl13anc 1325 . . . . 5 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑆𝑦)
2221ralrimiva 2960 . . . 4 (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
23 ssint 4460 . . . 4 (𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
2422, 23sylibr 224 . . 3 (𝜑𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
25 salgenval 39864 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
261, 25syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2724, 26sseqtr4d 3623 . 2 (𝜑𝑆 ⊆ (SalGen‘𝑋))
286, 27eqssd 3601 1 (𝜑 → (SalGen‘𝑋) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  wss 3556   cuni 4404   cint 4442  cfv 5849  SAlgcsalg 39851  SalGencsalgen 39855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-iota 5812  df-fun 5851  df-fv 5857  df-salg 39852  df-salgen 39856
This theorem is referenced by:  dfsalgen2  39882
  Copyright terms: Public domain W3C validator