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Theorem dfsalgen2 40877
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 2657 . . . . . . 7 ((SalGen‘𝑋) = 𝑆𝑆 = (SalGen‘𝑋))
32adantl 481 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4 dfsalgen2.1 . . . . . . . 8 (𝜑𝑋𝑉)
5 salgencl 40868 . . . . . . . 8 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
64, 5syl 17 . . . . . . 7 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
76adantr 480 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
83, 7eqeltrd 2730 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9 unieq 4476 . . . . . . 7 ((SalGen‘𝑋) = 𝑆 (SalGen‘𝑋) = 𝑆)
109adantl 481 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
114adantr 480 . . . . . . 7 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑉)
12 eqid 2651 . . . . . . 7 (SalGen‘𝑋) = (SalGen‘𝑋)
13 eqid 2651 . . . . . . 7 𝑋 = 𝑋
1411, 12, 13salgenuni 40873 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑋)
1510, 14eqtr3d 2687 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = 𝑋)
1612sssalgen 40871 . . . . . . 7 (𝑋𝑉𝑋 ⊆ (SalGen‘𝑋))
1711, 16syl 17 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18 simpr 476 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
1917, 18sseqtrd 3674 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑆)
208, 15, 193jca 1261 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆))
213ad2antrr 762 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑆 = (SalGen‘𝑋))
2221adantrl 752 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆 = (SalGen‘𝑋))
2311ad2antrr 762 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑉)
2423adantrl 752 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑉)
25 simplr 807 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑦 ∈ SAlg)
2625adantrl 752 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 ∈ SAlg)
27 simpr 476 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑦)
2827adantrl 752 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑦)
29 simprl 809 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 = 𝑋)
3024, 12, 26, 28, 29salgenss 40872 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
3122, 30eqsstrd 3672 . . . . . 6 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
3231ex 449 . . . . 5 (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3332ralrimiva 2995 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3420, 33jca 553 . . 3 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)))
3534ex 449 . 2 (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
364adantr 480 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑉)
37 simprl1 1126 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 ∈ SAlg)
38 simprl2 1127 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 = 𝑋)
39 simprl3 1128 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑆)
40 unieq 4476 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 𝑦 = 𝑤)
4140eqeq1d 2653 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → ( 𝑦 = 𝑋 𝑤 = 𝑋))
42 sseq2 3660 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑋𝑦𝑋𝑤))
4341, 42anbi12d 747 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (( 𝑦 = 𝑋𝑋𝑦) ↔ ( 𝑤 = 𝑋𝑋𝑤)))
44 sseq2 3660 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑆𝑦𝑆𝑤))
4543, 44imbi12d 333 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤)))
4645cbvralv 3201 . . . . . . . . . . 11 (∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
4746biimpi 206 . . . . . . . . . 10 (∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
4847adantr 480 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
49 simpr 476 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
5048, 49jca 553 . . . . . . . 8 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
51503ad2antr1 1246 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
52 3simpc 1080 . . . . . . . 8 ((𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤) → ( 𝑤 = 𝑋𝑋𝑤))
5352adantl 481 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → ( 𝑤 = 𝑋𝑋𝑤))
54 rspa 2959 . . . . . . 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 750 . . . . 5 ((((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5756adantll 750 . . . 4 (((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5836, 37, 38, 39, 57issalgend 40874 . . 3 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → (SalGen‘𝑋) = 𝑆)
5958ex 449 . 2 (𝜑 → (((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) → (SalGen‘𝑋) = 𝑆))
6035, 59impbid 202 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wss 3607   cuni 4468  cfv 5926  SAlgcsalg 40846  SalGencsalgen 40850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-salg 40847  df-salgen 40851
This theorem is referenced by:  unisalgen2  40890
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