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Theorem ldsysgenld 31840
Description: The intersection of all lambda-systems containing a given collection of sets 𝐴, which is called the lambda-system generated by 𝐴, is itself also a lambda-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)
Hypotheses
Ref Expression
isldsys.l 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
ldsysgenld.1 (𝜑𝑂𝑉)
ldsysgenld.2 (𝜑𝐴 ⊆ 𝒫 𝑂)
Assertion
Ref Expression
ldsysgenld (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
Distinct variable groups:   𝑦,𝑠   𝑡,𝐿   𝑂,𝑠,𝑡,𝑥   𝑥,𝑉   𝑦,𝑡   𝐴,𝑠,𝑡,𝑥   𝐿,𝑠,𝑥   𝜑,𝑡,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑠)   𝐴(𝑦)   𝐿(𝑦)   𝑂(𝑦)   𝑉(𝑦,𝑡,𝑠)

Proof of Theorem ldsysgenld
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ldsysgenld.1 . . . . 5 (𝜑𝑂𝑉)
2 pwsiga 31810 . . . . 5 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
31, 2syl 17 . . . 4 (𝜑 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
4 isldsys.l . . . . . . . 8 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥𝑠 (𝑂𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑠))}
54sigaldsys 31839 . . . . . . 7 (sigAlgebra‘𝑂) ⊆ 𝐿
65, 3sseldi 3899 . . . . . 6 (𝜑 → 𝒫 𝑂𝐿)
7 ldsysgenld.2 . . . . . 6 (𝜑𝐴 ⊆ 𝒫 𝑂)
8 sseq2 3927 . . . . . . 7 (𝑡 = 𝒫 𝑂 → (𝐴𝑡𝐴 ⊆ 𝒫 𝑂))
98elrab 3602 . . . . . 6 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} ↔ (𝒫 𝑂𝐿𝐴 ⊆ 𝒫 𝑂))
106, 7, 9sylanbrc 586 . . . . 5 (𝜑 → 𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡})
11 intss1 4874 . . . . 5 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} → {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
1210, 11syl 17 . . . 4 (𝜑 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
133, 12sselpwd 5219 . . 3 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂)
144isldsys 31836 . . . . . . . . . 10 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
1514simprbi 500 . . . . . . . . 9 (𝑡𝐿 → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
1615simp1d 1144 . . . . . . . 8 (𝑡𝐿 → ∅ ∈ 𝑡)
1716adantl 485 . . . . . . 7 ((𝜑𝑡𝐿) → ∅ ∈ 𝑡)
1817a1d 25 . . . . . 6 ((𝜑𝑡𝐿) → (𝐴𝑡 → ∅ ∈ 𝑡))
1918ralrimiva 3105 . . . . 5 (𝜑 → ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
20 0ex 5200 . . . . . 6 ∅ ∈ V
2120elintrab 4871 . . . . 5 (∅ ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
2219, 21sylibr 237 . . . 4 (𝜑 → ∅ ∈ {𝑡𝐿𝐴𝑡})
23 nfv 1922 . . . . . . . 8 𝑡𝜑
24 nfcv 2904 . . . . . . . . 9 𝑡𝑥
25 nfrab1 3296 . . . . . . . . . 10 𝑡{𝑡𝐿𝐴𝑡}
2625nfint 4869 . . . . . . . . 9 𝑡 {𝑡𝐿𝐴𝑡}
2724, 26nfel 2918 . . . . . . . 8 𝑡 𝑥 {𝑡𝐿𝐴𝑡}
2823, 27nfan 1907 . . . . . . 7 𝑡(𝜑𝑥 {𝑡𝐿𝐴𝑡})
29 simplr 769 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
30 vex 3412 . . . . . . . . . . . . . . 15 𝑥 ∈ V
3130elintrab 4871 . . . . . . . . . . . . . 14 (𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3231biimpi 219 . . . . . . . . . . . . 13 (𝑥 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3332adantl 485 . . . . . . . . . . . 12 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3433r19.21bi 3130 . . . . . . . . . . 11 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡𝑥𝑡))
3534imp 410 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
3615simp2d 1145 . . . . . . . . . . 11 (𝑡𝐿 → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
3736r19.21bi 3130 . . . . . . . . . 10 ((𝑡𝐿𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
3829, 35, 37syl2anc 587 . . . . . . . . 9 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑂𝑥) ∈ 𝑡)
3938ex 416 . . . . . . . 8 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
4039ex 416 . . . . . . 7 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑡𝐿 → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4128, 40ralrimi 3137 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
42 difexg 5220 . . . . . . . 8 (𝑂𝑉 → (𝑂𝑥) ∈ V)
43 elintrabg 4872 . . . . . . . 8 ((𝑂𝑥) ∈ V → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
441, 42, 433syl 18 . . . . . . 7 (𝜑 → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4544adantr 484 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4641, 45mpbird 260 . . . . 5 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4746ralrimiva 3105 . . . 4 (𝜑 → ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4826nfpw 4534 . . . . . . . . . . 11 𝑡𝒫 {𝑡𝐿𝐴𝑡}
4924, 48nfel 2918 . . . . . . . . . 10 𝑡 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}
5023, 49nfan 1907 . . . . . . . . 9 𝑡(𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
51 nfv 1922 . . . . . . . . 9 𝑡(𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)
5250, 51nfan 1907 . . . . . . . 8 𝑡((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
53 simplr 769 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
54 simpr 488 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢 {𝑡𝐿𝐴𝑡})
55 simpllr 776 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑡𝐿)
56 simplr 769 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝐴𝑡)
57 vex 3412 . . . . . . . . . . . . . . . . . . . 20 𝑢 ∈ V
5857elintrab 4871 . . . . . . . . . . . . . . . . . . 19 (𝑢 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
5958biimpi 219 . . . . . . . . . . . . . . . . . 18 (𝑢 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
6059r19.21bi 3130 . . . . . . . . . . . . . . . . 17 ((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) → (𝐴𝑡𝑢𝑡))
6160imp 410 . . . . . . . . . . . . . . . 16 (((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑢𝑡)
6254, 55, 56, 61syl21anc 838 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢𝑡)
6362ex 416 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑢 {𝑡𝐿𝐴𝑡} → 𝑢𝑡))
6463ssrdv 3907 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → {𝑡𝐿𝐴𝑡} ⊆ 𝑡)
6564sspwd 4528 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝒫 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑡)
66 simp-4r 784 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
6765, 66sseldd 3902 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 𝑡)
68 simpllr 776 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
6915simp3d 1146 . . . . . . . . . . . . 13 (𝑡𝐿 → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7069r19.21bi 3130 . . . . . . . . . . . 12 ((𝑡𝐿𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7170imp 410 . . . . . . . . . . 11 (((𝑡𝐿𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
7253, 67, 68, 71syl21anc 838 . . . . . . . . . 10 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
7372ex 416 . . . . . . . . 9 ((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) → (𝐴𝑡 𝑥𝑡))
7473ex 416 . . . . . . . 8 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑡𝐿 → (𝐴𝑡 𝑥𝑡)))
7552, 74ralrimi 3137 . . . . . . 7 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
76 vuniex 7527 . . . . . . . 8 𝑥 ∈ V
7776elintrab 4871 . . . . . . 7 ( 𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
7875, 77sylibr 237 . . . . . 6 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 {𝑡𝐿𝐴𝑡})
7978ex 416 . . . . 5 ((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8079ralrimiva 3105 . . . 4 (𝜑 → ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8122, 47, 803jca 1130 . . 3 (𝜑 → (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡})))
8213, 81jca 515 . 2 (𝜑 → ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
834isldsys 31836 . 2 ( {𝑡𝐿𝐴𝑡} ∈ 𝐿 ↔ ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
8482, 83sylibr 237 1 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  {crab 3065  Vcvv 3408  cdif 3863  wss 3866  c0 4237  𝒫 cpw 4513   cuni 4819   cint 4859  Disj wdisj 5018   class class class wbr 5053  cfv 6380  ωcom 7644  cdom 8624  sigAlgebracsiga 31788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-ac2 10077
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-oi 9126  df-dju 9517  df-card 9555  df-acn 9558  df-ac 9730  df-siga 31789
This theorem is referenced by:  ldgenpisys  31846  dynkin  31847
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