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Theorem issalgend 40874
Description: One side of dfsalgen2 40877. 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 2651 . . 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 40872 . 2 (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7 simpl 472 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝜑)
8 elrabi 3391 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 ∈ SAlg)
98adantl 481 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 ∈ SAlg)
10 unieq 4476 . . . . . . . . . . . 12 (𝑠 = 𝑦 𝑠 = 𝑦)
1110eqeq1d 2653 . . . . . . . . . . 11 (𝑠 = 𝑦 → ( 𝑠 = 𝑋 𝑦 = 𝑋))
12 sseq2 3660 . . . . . . . . . . 11 (𝑠 = 𝑦 → (𝑋𝑠𝑋𝑦))
1311, 12anbi12d 747 . . . . . . . . . 10 (𝑠 = 𝑦 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑦 = 𝑋𝑋𝑦)))
1413elrab 3396 . . . . . . . . 9 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1514biimpi 206 . . . . . . . 8 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1615simprld 810 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 = 𝑋)
1716adantl 481 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 = 𝑋)
1815simprrd 812 . . . . . . 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 1368 . . . . 5 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑆𝑦)
2221ralrimiva 2995 . . . 4 (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
23 ssint 4525 . . . 4 (𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
2422, 23sylibr 224 . . 3 (𝜑𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
25 salgenval 40859 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
261, 25syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2724, 26sseqtr4d 3675 . 2 (𝜑𝑆 ⊆ (SalGen‘𝑋))
286, 27eqssd 3653 1 (𝜑 → (SalGen‘𝑋) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  wss 3607   cuni 4468   cint 4507  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:  dfsalgen2  40877
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