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Theorem ldsysgenld 32759
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 32729 . . . . 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 32758 . . . . . . 7 (sigAlgebra‘𝑂) ⊆ 𝐿
65, 3sselid 3942 . . . . . 6 (𝜑 → 𝒫 𝑂𝐿)
7 ldsysgenld.2 . . . . . 6 (𝜑𝐴 ⊆ 𝒫 𝑂)
8 sseq2 3970 . . . . . . 7 (𝑡 = 𝒫 𝑂 → (𝐴𝑡𝐴 ⊆ 𝒫 𝑂))
98elrab 3645 . . . . . 6 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} ↔ (𝒫 𝑂𝐿𝐴 ⊆ 𝒫 𝑂))
106, 7, 9sylanbrc 583 . . . . 5 (𝜑 → 𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡})
11 intss1 4924 . . . . 5 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} → {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
1210, 11syl 17 . . . 4 (𝜑 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
133, 12sselpwd 5283 . . 3 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂)
144isldsys 32755 . . . . . . . . . 10 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
1514simprbi 497 . . . . . . . . 9 (𝑡𝐿 → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
1615simp1d 1142 . . . . . . . 8 (𝑡𝐿 → ∅ ∈ 𝑡)
1716adantl 482 . . . . . . 7 ((𝜑𝑡𝐿) → ∅ ∈ 𝑡)
1817a1d 25 . . . . . 6 ((𝜑𝑡𝐿) → (𝐴𝑡 → ∅ ∈ 𝑡))
1918ralrimiva 3143 . . . . 5 (𝜑 → ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
20 0ex 5264 . . . . . 6 ∅ ∈ V
2120elintrab 4921 . . . . 5 (∅ ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
2219, 21sylibr 233 . . . 4 (𝜑 → ∅ ∈ {𝑡𝐿𝐴𝑡})
23 nfv 1917 . . . . . . . 8 𝑡𝜑
24 nfcv 2907 . . . . . . . . 9 𝑡𝑥
25 nfrab1 3426 . . . . . . . . . 10 𝑡{𝑡𝐿𝐴𝑡}
2625nfint 4917 . . . . . . . . 9 𝑡 {𝑡𝐿𝐴𝑡}
2724, 26nfel 2921 . . . . . . . 8 𝑡 𝑥 {𝑡𝐿𝐴𝑡}
2823, 27nfan 1902 . . . . . . 7 𝑡(𝜑𝑥 {𝑡𝐿𝐴𝑡})
29 simplr 767 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
30 vex 3449 . . . . . . . . . . . . . . 15 𝑥 ∈ V
3130elintrab 4921 . . . . . . . . . . . . . 14 (𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3231biimpi 215 . . . . . . . . . . . . 13 (𝑥 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3332adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3433r19.21bi 3234 . . . . . . . . . . 11 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡𝑥𝑡))
3534imp 407 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
3615simp2d 1143 . . . . . . . . . . 11 (𝑡𝐿 → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
3736r19.21bi 3234 . . . . . . . . . 10 ((𝑡𝐿𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
3829, 35, 37syl2anc 584 . . . . . . . . 9 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑂𝑥) ∈ 𝑡)
3938ex 413 . . . . . . . 8 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
4039ex 413 . . . . . . 7 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑡𝐿 → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4128, 40ralrimi 3240 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
42 difexg 5284 . . . . . . . 8 (𝑂𝑉 → (𝑂𝑥) ∈ V)
43 elintrabg 4922 . . . . . . . 8 ((𝑂𝑥) ∈ V → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
441, 42, 433syl 18 . . . . . . 7 (𝜑 → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4544adantr 481 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ((𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4641, 45mpbird 256 . . . . 5 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4746ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4826nfpw 4579 . . . . . . . . . . 11 𝑡𝒫 {𝑡𝐿𝐴𝑡}
4924, 48nfel 2921 . . . . . . . . . 10 𝑡 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}
5023, 49nfan 1902 . . . . . . . . 9 𝑡(𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
51 nfv 1917 . . . . . . . . 9 𝑡(𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)
5250, 51nfan 1902 . . . . . . . 8 𝑡((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
53 simplr 767 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
54 simpr 485 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢 {𝑡𝐿𝐴𝑡})
55 simpllr 774 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑡𝐿)
56 simplr 767 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝐴𝑡)
57 vex 3449 . . . . . . . . . . . . . . . . . . . 20 𝑢 ∈ V
5857elintrab 4921 . . . . . . . . . . . . . . . . . . 19 (𝑢 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
5958biimpi 215 . . . . . . . . . . . . . . . . . 18 (𝑢 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
6059r19.21bi 3234 . . . . . . . . . . . . . . . . 17 ((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) → (𝐴𝑡𝑢𝑡))
6160imp 407 . . . . . . . . . . . . . . . 16 (((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑢𝑡)
6254, 55, 56, 61syl21anc 836 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢𝑡)
6362ex 413 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑢 {𝑡𝐿𝐴𝑡} → 𝑢𝑡))
6463ssrdv 3950 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → {𝑡𝐿𝐴𝑡} ⊆ 𝑡)
6564sspwd 4573 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝒫 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑡)
66 simp-4r 782 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
6765, 66sseldd 3945 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 𝑡)
68 simpllr 774 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
6915simp3d 1144 . . . . . . . . . . . . 13 (𝑡𝐿 → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7069r19.21bi 3234 . . . . . . . . . . . 12 ((𝑡𝐿𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7170imp 407 . . . . . . . . . . 11 (((𝑡𝐿𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
7253, 67, 68, 71syl21anc 836 . . . . . . . . . 10 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
7372ex 413 . . . . . . . . 9 ((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) → (𝐴𝑡 𝑥𝑡))
7473ex 413 . . . . . . . 8 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑡𝐿 → (𝐴𝑡 𝑥𝑡)))
7552, 74ralrimi 3240 . . . . . . 7 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
76 vuniex 7676 . . . . . . . 8 𝑥 ∈ V
7776elintrab 4921 . . . . . . 7 ( 𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
7875, 77sylibr 233 . . . . . 6 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 {𝑡𝐿𝐴𝑡})
7978ex 413 . . . . 5 ((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8079ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8122, 47, 803jca 1128 . . 3 (𝜑 → (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡})))
8213, 81jca 512 . 2 (𝜑 → ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
834isldsys 32755 . 2 ( {𝑡𝐿𝐴𝑡} ∈ 𝐿 ↔ ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
8482, 83sylibr 233 1 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865   cint 4907  Disj wdisj 5070   class class class wbr 5105  cfv 6496  ωcom 7802  cdom 8881  sigAlgebracsiga 32707
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-ac2 10399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9446  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-siga 32708
This theorem is referenced by:  ldgenpisys  32765  dynkin  32766
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