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Theorem issalgend 45054
Description: One side of dfsalgen2 45057. 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 2733 . . 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 45052 . 2 (πœ‘ β†’ (SalGenβ€˜π‘‹) βŠ† 𝑆)
7 simpl 484 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ πœ‘)
8 elrabi 3678 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ 𝑦 ∈ SAlg)
98adantl 483 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ 𝑦 ∈ SAlg)
10 unieq 4920 . . . . . . . . . . . 12 (𝑠 = 𝑦 β†’ βˆͺ 𝑠 = βˆͺ 𝑦)
1110eqeq1d 2735 . . . . . . . . . . 11 (𝑠 = 𝑦 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑦 = βˆͺ 𝑋))
12 sseq2 4009 . . . . . . . . . . 11 (𝑠 = 𝑦 β†’ (𝑋 βŠ† 𝑠 ↔ 𝑋 βŠ† 𝑦))
1311, 12anbi12d 632 . . . . . . . . . 10 (𝑠 = 𝑦 β†’ ((βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠) ↔ (βˆͺ 𝑦 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑦)))
1413elrab 3684 . . . . . . . . 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 771 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ βˆͺ 𝑦 = βˆͺ 𝑋)
1716adantl 483 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ βˆͺ 𝑦 = βˆͺ 𝑋)
1815simprrd 773 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)} β†’ 𝑋 βŠ† 𝑦)
1918adantl 483 . . . . . 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 1373 . . . . 5 ((πœ‘ ∧ 𝑦 ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}) β†’ 𝑆 βŠ† 𝑦)
2221ralrimiva 3147 . . . 4 (πœ‘ β†’ βˆ€π‘¦ ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}𝑆 βŠ† 𝑦)
23 ssint 4969 . . . 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 45037 . . . 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 4024 . 2 (πœ‘ β†’ 𝑆 βŠ† (SalGenβ€˜π‘‹))
286, 27eqssd 4000 1 (πœ‘ β†’ (SalGenβ€˜π‘‹) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   βŠ† wss 3949  βˆͺ cuni 4909  βˆ© cint 4951  β€˜cfv 6544  SAlgcsalg 45024  SalGencsalgen 45028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-salg 45025  df-salgen 45029
This theorem is referenced by:  dfsalgen2  45057
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