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Theorem dfsalgen2 41076
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 2777 . . . . . . 7 ((SalGen‘𝑋) = 𝑆𝑆 = (SalGen‘𝑋))
32adantl 467 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4 dfsalgen2.1 . . . . . . . 8 (𝜑𝑋𝑉)
5 salgencl 41067 . . . . . . . 8 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
64, 5syl 17 . . . . . . 7 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
76adantr 466 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
83, 7eqeltrd 2850 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9 unieq 4582 . . . . . . 7 ((SalGen‘𝑋) = 𝑆 (SalGen‘𝑋) = 𝑆)
109adantl 467 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
114adantr 466 . . . . . . 7 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑉)
12 eqid 2771 . . . . . . 7 (SalGen‘𝑋) = (SalGen‘𝑋)
13 eqid 2771 . . . . . . 7 𝑋 = 𝑋
1411, 12, 13salgenuni 41072 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑋)
1510, 14eqtr3d 2807 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = 𝑋)
1612sssalgen 41070 . . . . . . 7 (𝑋𝑉𝑋 ⊆ (SalGen‘𝑋))
1711, 16syl 17 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18 simpr 471 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
1917, 18sseqtrd 3790 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑆)
208, 15, 193jca 1122 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆))
213ad2antrr 697 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑆 = (SalGen‘𝑋))
2221adantrl 687 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆 = (SalGen‘𝑋))
2311ad2antrr 697 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑉)
2423adantrl 687 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑉)
25 simplr 744 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑦 ∈ SAlg)
2625adantrl 687 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 ∈ SAlg)
27 simpr 471 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑦)
2827adantrl 687 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑦)
29 simprl 746 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 = 𝑋)
3024, 12, 26, 28, 29salgenss 41071 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
3122, 30eqsstrd 3788 . . . . . 6 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
3231ex 397 . . . . 5 (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3332ralrimiva 3115 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3420, 33jca 495 . . 3 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)))
3534ex 397 . 2 (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
364adantr 466 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑉)
37 simprl1 1266 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 ∈ SAlg)
38 simprl2 1268 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 = 𝑋)
39 simprl3 1270 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑆)
40 unieq 4582 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 𝑦 = 𝑤)
4140eqeq1d 2773 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → ( 𝑦 = 𝑋 𝑤 = 𝑋))
42 sseq2 3776 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑋𝑦𝑋𝑤))
4341, 42anbi12d 608 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (( 𝑦 = 𝑋𝑋𝑦) ↔ ( 𝑤 = 𝑋𝑋𝑤)))
44 sseq2 3776 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑆𝑦𝑆𝑤))
4543, 44imbi12d 333 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤)))
4645cbvralv 3320 . . . . . . . . . . 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 466 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
49 simpr 471 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
5048, 49jca 495 . . . . . . . 8 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
51503ad2antr1 1203 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
52 3simpc 1146 . . . . . . . 8 ((𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤) → ( 𝑤 = 𝑋𝑋𝑤))
5352adantl 467 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → ( 𝑤 = 𝑋𝑋𝑤))
54 rspa 3079 . . . . . . 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 685 . . . . 5 ((((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5756adantll 685 . . . 4 (((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5836, 37, 38, 39, 57issalgend 41073 . . 3 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → (SalGen‘𝑋) = 𝑆)
5958ex 397 . 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 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wss 3723   cuni 4574  cfv 6031  SAlgcsalg 41045  SalGencsalgen 41049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-salg 41046  df-salgen 41050
This theorem is referenced by:  unisalgen2  41089
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