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Theorem issalgend 44569
Description: One side of dfsalgen2 44572. 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 2736 . . 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 44567 . 2 (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7 simpl 483 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝜑)
8 elrabi 3639 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 ∈ SAlg)
98adantl 482 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 ∈ SAlg)
10 unieq 4876 . . . . . . . . . . . 12 (𝑠 = 𝑦 𝑠 = 𝑦)
1110eqeq1d 2738 . . . . . . . . . . 11 (𝑠 = 𝑦 → ( 𝑠 = 𝑋 𝑦 = 𝑋))
12 sseq2 3970 . . . . . . . . . . 11 (𝑠 = 𝑦 → (𝑋𝑠𝑋𝑦))
1311, 12anbi12d 631 . . . . . . . . . 10 (𝑠 = 𝑦 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑦 = 𝑋𝑋𝑦)))
1413elrab 3645 . . . . . . . . 9 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1514biimpi 215 . . . . . . . 8 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1615simprld 770 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 = 𝑋)
1716adantl 482 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 = 𝑋)
1815simprrd 772 . . . . . . 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 1372 . . . . 5 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑆𝑦)
2221ralrimiva 3143 . . . 4 (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
23 ssint 4925 . . . 4 (𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
2422, 23sylibr 233 . . 3 (𝜑𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
25 salgenval 44552 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
261, 25syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2724, 26sseqtrrd 3985 . 2 (𝜑𝑆 ⊆ (SalGen‘𝑋))
286, 27eqssd 3961 1 (𝜑 → (SalGen‘𝑋) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  wss 3910   cuni 4865   cint 4907  cfv 6496  SAlgcsalg 44539  SalGencsalgen 44543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-salg 44540  df-salgen 44544
This theorem is referenced by:  dfsalgen2  44572
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