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Theorem dfsalgen2 41196
Description: Alternate characterization of the sigma-algebra generated by a set. It is the smallest sigma-algebra, on the same base set, that includes the set. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
Hypothesis
Ref Expression
dfsalgen2.1 (𝜑𝑋𝑉)
Assertion
Ref Expression
dfsalgen2 (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
Distinct variable groups:   𝑦,𝑆   𝑦,𝑋   𝜑,𝑦
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfsalgen2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . . . . 8 ((SalGen‘𝑋) = 𝑆 → (SalGen‘𝑋) = 𝑆)
21eqcomd 2771 . . . . . . 7 ((SalGen‘𝑋) = 𝑆𝑆 = (SalGen‘𝑋))
32adantl 473 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4 dfsalgen2.1 . . . . . . . 8 (𝜑𝑋𝑉)
5 salgencl 41187 . . . . . . . 8 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
64, 5syl 17 . . . . . . 7 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
76adantr 472 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
83, 7eqeltrd 2844 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9 unieq 4602 . . . . . . 7 ((SalGen‘𝑋) = 𝑆 (SalGen‘𝑋) = 𝑆)
109adantl 473 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
114adantr 472 . . . . . . 7 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑉)
12 eqid 2765 . . . . . . 7 (SalGen‘𝑋) = (SalGen‘𝑋)
13 eqid 2765 . . . . . . 7 𝑋 = 𝑋
1411, 12, 13salgenuni 41192 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑋)
1510, 14eqtr3d 2801 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = 𝑋)
1612sssalgen 41190 . . . . . . 7 (𝑋𝑉𝑋 ⊆ (SalGen‘𝑋))
1711, 16syl 17 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18 simpr 477 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
1917, 18sseqtrd 3801 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑆)
208, 15, 193jca 1158 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆))
213ad2antrr 717 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑆 = (SalGen‘𝑋))
2221adantrl 707 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆 = (SalGen‘𝑋))
2311ad2antrr 717 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑉)
2423adantrl 707 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑉)
25 simplr 785 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑦 ∈ SAlg)
2625adantrl 707 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 ∈ SAlg)
27 simpr 477 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑦)
2827adantrl 707 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑦)
29 simprl 787 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 = 𝑋)
3024, 12, 26, 28, 29salgenss 41191 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
3122, 30eqsstrd 3799 . . . . . 6 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
3231ex 401 . . . . 5 (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3332ralrimiva 3113 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3420, 33jca 507 . . 3 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)))
3534ex 401 . 2 (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
364adantr 472 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑉)
37 simprl1 1281 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 ∈ SAlg)
38 simprl2 1283 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 = 𝑋)
39 simprl3 1285 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑆)
40 unieq 4602 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 𝑦 = 𝑤)
4140eqeq1d 2767 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → ( 𝑦 = 𝑋 𝑤 = 𝑋))
42 sseq2 3787 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑋𝑦𝑋𝑤))
4341, 42anbi12d 624 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (( 𝑦 = 𝑋𝑋𝑦) ↔ ( 𝑤 = 𝑋𝑋𝑤)))
44 sseq2 3787 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑆𝑦𝑆𝑤))
4543, 44imbi12d 335 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤)))
4645cbvralv 3319 . . . . . . . . . . 11 (∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
4746biimpi 207 . . . . . . . . . 10 (∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
4847adantr 472 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
49 simpr 477 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
5048, 49jca 507 . . . . . . . 8 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
51503ad2antr1 1239 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
52 3simpc 1182 . . . . . . . 8 ((𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤) → ( 𝑤 = 𝑋𝑋𝑤))
5352adantl 473 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → ( 𝑤 = 𝑋𝑋𝑤))
54 rspa 3077 . . . . . . 7 ((∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg) → (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
5551, 53, 54sylc 65 . . . . . 6 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5655adantll 705 . . . . 5 ((((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5756adantll 705 . . . 4 (((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5836, 37, 38, 39, 57issalgend 41193 . . 3 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → (SalGen‘𝑋) = 𝑆)
5958ex 401 . 2 (𝜑 → (((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) → (SalGen‘𝑋) = 𝑆))
6035, 59impbid 203 1 (𝜑 → ((SalGen‘𝑋) = 𝑆 ↔ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wss 3732   cuni 4594  cfv 6068  SAlgcsalg 41165  SalGencsalgen 41169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-int 4634  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-salg 41166  df-salgen 41170
This theorem is referenced by:  unisalgen2  41209
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