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Theorem issalgend 45133
Description: One side of dfsalgen2 45136. If a sigma-algebra on βˆͺ 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 (πœ‘ β†’ 𝑋 ∈ 𝑉)
issalgend.s (πœ‘ β†’ 𝑆 ∈ SAlg)
issalgend.u (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
issalgend.i (πœ‘ β†’ 𝑋 βŠ† 𝑆)
issalgend.a ((πœ‘ ∧ (𝑦 ∈ SAlg ∧ βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑆 βŠ† 𝑦)
Assertion
Ref Expression
issalgend (πœ‘ β†’ (SalGenβ€˜π‘‹) = 𝑆)
Distinct variable groups:   𝑦,𝑆   𝑦,𝑋   πœ‘,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 (πœ‘ β†’ 𝑋 ∈ 𝑉)
2 eqid 2732 . . 3 (SalGenβ€˜π‘‹) = (SalGenβ€˜π‘‹)
3 issalgend.s . . 3 (πœ‘ β†’ 𝑆 ∈ SAlg)
4 issalgend.i . . 3 (πœ‘ β†’ 𝑋 βŠ† 𝑆)
5 issalgend.u . . 3 (πœ‘ β†’ βˆͺ 𝑆 = βˆͺ 𝑋)
61, 2, 3, 4, 5salgenss 45131 . 2 (πœ‘ β†’ (SalGenβ€˜π‘‹) βŠ† 𝑆)
7 simpl 483 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ πœ‘)
8 elrabi 3677 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ 𝑦 ∈ SAlg)
98adantl 482 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ 𝑦 ∈ SAlg)
10 unieq 4919 . . . . . . . . . . . 12 (𝑠 = 𝑦 β†’ βˆͺ 𝑠 = βˆͺ 𝑦)
1110eqeq1d 2734 . . . . . . . . . . 11 (𝑠 = 𝑦 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑦 = βˆͺ 𝑋))
12 sseq2 4008 . . . . . . . . . . 11 (𝑠 = 𝑦 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑦))
1311, 12anbi12d 631 . . . . . . . . . 10 (𝑠 = 𝑦 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)))
1413elrab 3683 . . . . . . . . 9 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ↔ (𝑦 ∈ SAlg ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)))
1514biimpi 215 . . . . . . . 8 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ (𝑦 ∈ SAlg ∧ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)))
1615simprld 770 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑦 = βˆͺ 𝑋)
1716adantl 482 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ βˆͺ 𝑦 = βˆͺ 𝑋)
1815simprrd 772 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ 𝑋 βŠ† 𝑦)
1918adantl 482 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ 𝑋 βŠ† 𝑦)
20 issalgend.a . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ SAlg ∧ βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)) β†’ 𝑆 βŠ† 𝑦)
217, 9, 17, 19, 20syl13anc 1372 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ 𝑆 βŠ† 𝑦)
2221ralrimiva 3146 . . . 4 (πœ‘ β†’ βˆ€π‘¦ ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}𝑆 βŠ† 𝑦)
23 ssint 4968 . . . 4 (𝑆 βŠ† ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} ↔ βˆ€π‘¦ ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}𝑆 βŠ† 𝑦)
2422, 23sylibr 233 . . 3 (πœ‘ β†’ 𝑆 βŠ† ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
25 salgenval 45116 . . . 4 (𝑋 ∈ 𝑉 β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
261, 25syl 17 . . 3 (πœ‘ β†’ (SalGenβ€˜π‘‹) = ∩ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)})
2724, 26sseqtrrd 4023 . 2 (πœ‘ β†’ 𝑆 βŠ† (SalGenβ€˜π‘‹))
286, 27eqssd 3999 1 (πœ‘ β†’ (SalGenβ€˜π‘‹) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   βŠ† wss 3948  βˆͺ cuni 4908  βˆ© cint 4950  β€˜cfv 6543  SAlgcsalg 45103  SalGencsalgen 45107
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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-salg 45104  df-salgen 45108
This theorem is referenced by:  dfsalgen2  45136
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