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Theorem ldsysgenld 30022
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 29992 . . . . 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 30021 . . . . . . 7 (sigAlgebra‘𝑂) ⊆ 𝐿
65, 3sseldi 3585 . . . . . 6 (𝜑 → 𝒫 𝑂𝐿)
7 ldsysgenld.2 . . . . . 6 (𝜑𝐴 ⊆ 𝒫 𝑂)
8 sseq2 3611 . . . . . . 7 (𝑡 = 𝒫 𝑂 → (𝐴𝑡𝐴 ⊆ 𝒫 𝑂))
98elrab 3350 . . . . . 6 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} ↔ (𝒫 𝑂𝐿𝐴 ⊆ 𝒫 𝑂))
106, 7, 9sylanbrc 697 . . . . 5 (𝜑 → 𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡})
11 intss1 4462 . . . . 5 (𝒫 𝑂 ∈ {𝑡𝐿𝐴𝑡} → {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
1210, 11syl 17 . . . 4 (𝜑 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑂)
133, 12sselpwd 4772 . . 3 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂)
144isldsys 30018 . . . . . . . . . 10 (𝑡𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))))
1514simprbi 480 . . . . . . . . 9 (𝑡𝐿 → (∅ ∈ 𝑡 ∧ ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡)))
1615simp1d 1071 . . . . . . . 8 (𝑡𝐿 → ∅ ∈ 𝑡)
1716adantl 482 . . . . . . 7 ((𝜑𝑡𝐿) → ∅ ∈ 𝑡)
1817a1d 25 . . . . . 6 ((𝜑𝑡𝐿) → (𝐴𝑡 → ∅ ∈ 𝑡))
1918ralrimiva 2961 . . . . 5 (𝜑 → ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
20 0ex 4755 . . . . . 6 ∅ ∈ V
2120elintrab 4458 . . . . 5 (∅ ∈ {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 → ∅ ∈ 𝑡))
2219, 21sylibr 224 . . . 4 (𝜑 → ∅ ∈ {𝑡𝐿𝐴𝑡})
23 nfv 1840 . . . . . . . 8 𝑡𝜑
24 nfcv 2761 . . . . . . . . 9 𝑡𝑥
25 nfrab1 3114 . . . . . . . . . 10 𝑡{𝑡𝐿𝐴𝑡}
2625nfint 4456 . . . . . . . . 9 𝑡 {𝑡𝐿𝐴𝑡}
2724, 26nfel 2773 . . . . . . . 8 𝑡 𝑥 {𝑡𝐿𝐴𝑡}
2823, 27nfan 1825 . . . . . . 7 𝑡(𝜑𝑥 {𝑡𝐿𝐴𝑡})
29 simplr 791 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
30 vex 3192 . . . . . . . . . . . . . . 15 𝑥 ∈ V
3130elintrab 4458 . . . . . . . . . . . . . 14 (𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3231biimpi 206 . . . . . . . . . . . . 13 (𝑥 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3332adantl 482 . . . . . . . . . . . 12 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡𝑥𝑡))
3433r19.21bi 2927 . . . . . . . . . . 11 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡𝑥𝑡))
3534imp 445 . . . . . . . . . 10 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
3615simp2d 1072 . . . . . . . . . . 11 (𝑡𝐿 → ∀𝑥𝑡 (𝑂𝑥) ∈ 𝑡)
3736r19.21bi 2927 . . . . . . . . . 10 ((𝑡𝐿𝑥𝑡) → (𝑂𝑥) ∈ 𝑡)
3829, 35, 37syl2anc 692 . . . . . . . . 9 ((((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑂𝑥) ∈ 𝑡)
3938ex 450 . . . . . . . 8 (((𝜑𝑥 {𝑡𝐿𝐴𝑡}) ∧ 𝑡𝐿) → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
4039ex 450 . . . . . . 7 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑡𝐿 → (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡)))
4128, 40ralrimi 2952 . . . . . 6 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → ∀𝑡𝐿 (𝐴𝑡 → (𝑂𝑥) ∈ 𝑡))
42 difexg 4773 . . . . . . . 8 (𝑂𝑉 → (𝑂𝑥) ∈ V)
43 elintrabg 4459 . . . . . . . 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 247 . . . . 5 ((𝜑𝑥 {𝑡𝐿𝐴𝑡}) → (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4746ralrimiva 2961 . . . 4 (𝜑 → ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡})
4826nfpw 4148 . . . . . . . . . . 11 𝑡𝒫 {𝑡𝐿𝐴𝑡}
4924, 48nfel 2773 . . . . . . . . . 10 𝑡 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}
5023, 49nfan 1825 . . . . . . . . 9 𝑡(𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
51 nfv 1840 . . . . . . . . 9 𝑡(𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)
5250, 51nfan 1825 . . . . . . . 8 𝑡((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
53 simplr 791 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑡𝐿)
54 simpr 477 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢 {𝑡𝐿𝐴𝑡})
55 simpllr 798 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑡𝐿)
56 simplr 791 . . . . . . . . . . . . . . . 16 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝐴𝑡)
57 vex 3192 . . . . . . . . . . . . . . . . . . . 20 𝑢 ∈ V
5857elintrab 4458 . . . . . . . . . . . . . . . . . . 19 (𝑢 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
5958biimpi 206 . . . . . . . . . . . . . . . . . 18 (𝑢 {𝑡𝐿𝐴𝑡} → ∀𝑡𝐿 (𝐴𝑡𝑢𝑡))
6059r19.21bi 2927 . . . . . . . . . . . . . . . . 17 ((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) → (𝐴𝑡𝑢𝑡))
6160imp 445 . . . . . . . . . . . . . . . 16 (((𝑢 {𝑡𝐿𝐴𝑡} ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑢𝑡)
6254, 55, 56, 61syl21anc 1322 . . . . . . . . . . . . . . 15 ((((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) ∧ 𝑢 {𝑡𝐿𝐴𝑡}) → 𝑢𝑡)
6362ex 450 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑢 {𝑡𝐿𝐴𝑡} → 𝑢𝑡))
6463ssrdv 3593 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → {𝑡𝐿𝐴𝑡} ⊆ 𝑡)
65 sspwb 4883 . . . . . . . . . . . . 13 ( {𝑡𝐿𝐴𝑡} ⊆ 𝑡 ↔ 𝒫 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑡)
6664, 65sylib 208 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝒫 {𝑡𝐿𝐴𝑡} ⊆ 𝒫 𝑡)
67 simp-4r 806 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡})
6866, 67sseldd 3588 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥 ∈ 𝒫 𝑡)
69 simpllr 798 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦))
7015simp3d 1073 . . . . . . . . . . . . 13 (𝑡𝐿 → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7170r19.21bi 2927 . . . . . . . . . . . 12 ((𝑡𝐿𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥𝑡))
7271imp 445 . . . . . . . . . . 11 (((𝑡𝐿𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥𝑡)
7353, 68, 69, 72syl21anc 1322 . . . . . . . . . 10 (((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) ∧ 𝐴𝑡) → 𝑥𝑡)
7473ex 450 . . . . . . . . 9 ((((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) ∧ 𝑡𝐿) → (𝐴𝑡 𝑥𝑡))
7574ex 450 . . . . . . . 8 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → (𝑡𝐿 → (𝐴𝑡 𝑥𝑡)))
7652, 75ralrimi 2952 . . . . . . 7 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
77 vuniex 6914 . . . . . . . 8 𝑥 ∈ V
7877elintrab 4458 . . . . . . 7 ( 𝑥 {𝑡𝐿𝐴𝑡} ↔ ∀𝑡𝐿 (𝐴𝑡 𝑥𝑡))
7976, 78sylibr 224 . . . . . 6 (((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦)) → 𝑥 {𝑡𝐿𝐴𝑡})
8079ex 450 . . . . 5 ((𝜑𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡}) → ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8180ralrimiva 2961 . . . 4 (𝜑 → ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))
8222, 47, 813jca 1240 . . 3 (𝜑 → (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡})))
8313, 82jca 554 . 2 (𝜑 → ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
844isldsys 30018 . 2 ( {𝑡𝐿𝐴𝑡} ∈ 𝐿 ↔ ( {𝑡𝐿𝐴𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 {𝑡𝐿𝐴𝑡} (𝑂𝑥) ∈ {𝑡𝐿𝐴𝑡} ∧ ∀𝑥 ∈ 𝒫 {𝑡𝐿𝐴𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦𝑥 𝑦) → 𝑥 {𝑡𝐿𝐴𝑡}))))
8583, 84sylibr 224 1 (𝜑 {𝑡𝐿𝐴𝑡} ∈ 𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3189  cdif 3556  wss 3559  c0 3896  𝒫 cpw 4135   cuni 4407   cint 4445  Disj wdisj 4588   class class class wbr 4618  cfv 5852  ωcom 7019  cdom 7904  sigAlgebracsiga 29969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-ac2 9236
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-oi 8366  df-card 8716  df-acn 8719  df-ac 8890  df-cda 8941  df-siga 29970
This theorem is referenced by:  ldgenpisys  30028  dynkin  30029
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