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Theorem dfsalgen2 42974
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 2807 . . . . . . 7 ((SalGen‘𝑋) = 𝑆𝑆 = (SalGen‘𝑋))
32adantl 485 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = (SalGen‘𝑋))
4 dfsalgen2.1 . . . . . . . 8 (𝜑𝑋𝑉)
5 salgencl 42965 . . . . . . . 8 (𝑋𝑉 → (SalGen‘𝑋) ∈ SAlg)
64, 5syl 17 . . . . . . 7 (𝜑 → (SalGen‘𝑋) ∈ SAlg)
76adantr 484 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) ∈ SAlg)
83, 7eqeltrd 2893 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 ∈ SAlg)
9 unieq 4814 . . . . . . 7 ((SalGen‘𝑋) = 𝑆 (SalGen‘𝑋) = 𝑆)
109adantl 485 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
114adantr 484 . . . . . . 7 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑉)
12 eqid 2801 . . . . . . 7 (SalGen‘𝑋) = (SalGen‘𝑋)
13 eqid 2801 . . . . . . 7 𝑋 = 𝑋
1411, 12, 13salgenuni 42970 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑋)
1510, 14eqtr3d 2838 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑆 = 𝑋)
1612sssalgen 42968 . . . . . . 7 (𝑋𝑉𝑋 ⊆ (SalGen‘𝑋))
1711, 16syl 17 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋 ⊆ (SalGen‘𝑋))
18 simpr 488 . . . . . 6 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (SalGen‘𝑋) = 𝑆)
1917, 18sseqtrd 3958 . . . . 5 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → 𝑋𝑆)
208, 15, 193jca 1125 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → (𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆))
213ad2antrr 725 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑆 = (SalGen‘𝑋))
2221adantrl 715 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆 = (SalGen‘𝑋))
2311ad2antrr 725 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑉)
2423adantrl 715 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑉)
25 simplr 768 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑦 ∈ SAlg)
2625adantrl 715 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 ∈ SAlg)
27 simpr 488 . . . . . . . . 9 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ 𝑋𝑦) → 𝑋𝑦)
2827adantrl 715 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑋𝑦)
29 simprl 770 . . . . . . . 8 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑦 = 𝑋)
3024, 12, 26, 28, 29salgenss 42969 . . . . . . 7 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → (SalGen‘𝑋) ⊆ 𝑦)
3122, 30eqsstrd 3956 . . . . . 6 ((((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) ∧ ( 𝑦 = 𝑋𝑋𝑦)) → 𝑆𝑦)
3231ex 416 . . . . 5 (((𝜑 ∧ (SalGen‘𝑋) = 𝑆) ∧ 𝑦 ∈ SAlg) → (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3332ralrimiva 3152 . . . 4 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))
3420, 33jca 515 . . 3 ((𝜑 ∧ (SalGen‘𝑋) = 𝑆) → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)))
3534ex 416 . 2 (𝜑 → ((SalGen‘𝑋) = 𝑆 → ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))))
364adantr 484 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑉)
37 simprl1 1215 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 ∈ SAlg)
38 simprl2 1216 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑆 = 𝑋)
39 simprl3 1217 . . . 4 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → 𝑋𝑆)
40 unieq 4814 . . . . . . . . . . . . . . 15 (𝑦 = 𝑤 𝑦 = 𝑤)
4140eqeq1d 2803 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → ( 𝑦 = 𝑋 𝑤 = 𝑋))
42 sseq2 3944 . . . . . . . . . . . . . 14 (𝑦 = 𝑤 → (𝑋𝑦𝑋𝑤))
4341, 42anbi12d 633 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (( 𝑦 = 𝑋𝑋𝑦) ↔ ( 𝑤 = 𝑋𝑋𝑤)))
44 sseq2 3944 . . . . . . . . . . . . 13 (𝑦 = 𝑤 → (𝑆𝑦𝑆𝑤))
4543, 44imbi12d 348 . . . . . . . . . . . 12 (𝑦 = 𝑤 → ((( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤)))
4645cbvralvw 3399 . . . . . . . . . . 11 (∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ↔ ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
4746biimpi 219 . . . . . . . . . 10 (∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
4847adantr 484 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → ∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤))
49 simpr 488 . . . . . . . . 9 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → 𝑤 ∈ SAlg)
5048, 49jca 515 . . . . . . . 8 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ 𝑤 ∈ SAlg) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
51503ad2antr1 1185 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → (∀𝑤 ∈ SAlg (( 𝑤 = 𝑋𝑋𝑤) → 𝑆𝑤) ∧ 𝑤 ∈ SAlg))
52 3simpc 1147 . . . . . . . 8 ((𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤) → ( 𝑤 = 𝑋𝑋𝑤))
5352adantl 485 . . . . . . 7 ((∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → ( 𝑤 = 𝑋𝑋𝑤))
54 rspa 3174 . . . . . . 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 713 . . . . 5 ((((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5756adantll 713 . . . 4 (((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) ∧ (𝑤 ∈ SAlg ∧ 𝑤 = 𝑋𝑋𝑤)) → 𝑆𝑤)
5836, 37, 38, 39, 57issalgend 42971 . . 3 ((𝜑 ∧ ((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦))) → (SalGen‘𝑋) = 𝑆)
5958ex 416 . 2 (𝜑 → (((𝑆 ∈ SAlg ∧ 𝑆 = 𝑋𝑋𝑆) ∧ ∀𝑦 ∈ SAlg (( 𝑦 = 𝑋𝑋𝑦) → 𝑆𝑦)) → (SalGen‘𝑋) = 𝑆))
6035, 59impbid 215 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wss 3884   cuni 4803  cfv 6328  SAlgcsalg 42943  SalGencsalgen 42947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-int 4842  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-salg 42944  df-salgen 42948
This theorem is referenced by:  unisalgen2  42987
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