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Theorem issalgend 44669
Description: One side of dfsalgen2 44672. 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 44667 . 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 3643 . . . . . . 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 4880 . . . . . . . . . . . 12 (𝑠 = 𝑦 β†’ βˆͺ 𝑠 = βˆͺ 𝑦)
1110eqeq1d 2734 . . . . . . . . . . 11 (𝑠 = 𝑦 β†’ (βˆͺ 𝑠 = βˆͺ 𝑋 ↔ βˆͺ 𝑦 = βˆͺ 𝑋))
12 sseq2 3974 . . . . . . . . . . 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 3649 . . . . . . . . 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 3140 . . . 4 (πœ‘ β†’ βˆ€π‘¦ ∈ {𝑠 ∈ SAlg ∣ (βˆͺ 𝑠 = βˆͺ 𝑋 ∧ 𝑋 βŠ† 𝑠)}𝑆 βŠ† 𝑦)
23 ssint 4929 . . . 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 44652 . . . 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 3989 . 2 (πœ‘ β†’ 𝑆 βŠ† (SalGenβ€˜π‘‹))
286, 27eqssd 3965 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 3061  {crab 3406   βŠ† wss 3914  βˆͺ cuni 4869  βˆ© cint 4911  β€˜cfv 6500  SAlgcsalg 44639  SalGencsalgen 44643
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 2703  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 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 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-salg 44640  df-salgen 44644
This theorem is referenced by:  dfsalgen2  44672
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