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Theorem issalgend 41303
Description: One side of dfsalgen2 41306. 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 2803 . . 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 41301 . 2 (𝜑 → (SalGen‘𝑋) ⊆ 𝑆)
7 simpl 475 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝜑)
8 elrabi 3555 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 ∈ SAlg)
98adantl 474 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 ∈ SAlg)
10 unieq 4640 . . . . . . . . . . . 12 (𝑠 = 𝑦 𝑠 = 𝑦)
1110eqeq1d 2805 . . . . . . . . . . 11 (𝑠 = 𝑦 → ( 𝑠 = 𝑋 𝑦 = 𝑋))
12 sseq2 3827 . . . . . . . . . . 11 (𝑠 = 𝑦 → (𝑋𝑠𝑋𝑦))
1311, 12anbi12d 625 . . . . . . . . . 10 (𝑠 = 𝑦 → (( 𝑠 = 𝑋𝑋𝑠) ↔ ( 𝑦 = 𝑋𝑋𝑦)))
1413elrab 3560 . . . . . . . . 9 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1514biimpi 208 . . . . . . . 8 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → (𝑦 ∈ SAlg ∧ ( 𝑦 = 𝑋𝑋𝑦)))
1615simprld 789 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑦 = 𝑋)
1716adantl 474 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑦 = 𝑋)
1815simprrd 791 . . . . . . 7 (𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} → 𝑋𝑦)
1918adantl 474 . . . . . 6 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑋𝑦)
20 issalgend.a . . . . . 6 ((𝜑 ∧ (𝑦 ∈ SAlg ∧ 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
217, 9, 17, 19, 20syl13anc 1492 . . . . 5 ((𝜑𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}) → 𝑆𝑦)
2221ralrimiva 3151 . . . 4 (𝜑 → ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
23 ssint 4687 . . . 4 (𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)} ↔ ∀𝑦 ∈ {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)}𝑆𝑦)
2422, 23sylibr 226 . . 3 (𝜑𝑆 {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
25 salgenval 41288 . . . 4 (𝑋𝑉 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
261, 25syl 17 . . 3 (𝜑 → (SalGen‘𝑋) = {𝑠 ∈ SAlg ∣ ( 𝑠 = 𝑋𝑋𝑠)})
2724, 26sseqtr4d 3842 . 2 (𝜑𝑆 ⊆ (SalGen‘𝑋))
286, 27eqssd 3819 1 (𝜑 → (SalGen‘𝑋) = 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wral 3093  {crab 3097  wss 3773   cuni 4632   cint 4671  cfv 6105  SAlgcsalg 41275  SalGencsalgen 41279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2379  ax-ext 2781  ax-sep 4979  ax-nul 4987  ax-pow 5039  ax-pr 5101  ax-un 7187
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2593  df-eu 2611  df-clab 2790  df-cleq 2796  df-clel 2799  df-nfc 2934  df-ne 2976  df-ral 3098  df-rex 3099  df-rab 3102  df-v 3391  df-sbc 3638  df-csb 3733  df-dif 3776  df-un 3778  df-in 3780  df-ss 3787  df-nul 4120  df-if 4282  df-pw 4355  df-sn 4373  df-pr 4375  df-op 4379  df-uni 4633  df-int 4672  df-br 4848  df-opab 4910  df-mpt 4927  df-id 5224  df-xp 5322  df-rel 5323  df-cnv 5324  df-co 5325  df-dm 5326  df-iota 6068  df-fun 6107  df-fv 6113  df-salg 41276  df-salgen 41280
This theorem is referenced by:  dfsalgen2  41306
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