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Theorem subsaliuncllem 46372
Description: A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
subsaliuncllem.f 𝑦𝜑
subsaliuncllem.s (𝜑𝑆𝑉)
subsaliuncllem.g 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
subsaliuncllem.e 𝐸 = (𝐻𝐺)
subsaliuncllem.h (𝜑𝐻 Fn ran 𝐺)
subsaliuncllem.y (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
Assertion
Ref Expression
subsaliuncllem (𝜑 → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
Distinct variable groups:   𝐷,𝑒   𝑥,𝐷   𝑒,𝐸,𝑛   𝑥,𝐸,𝑛   𝑒,𝐹   𝑥,𝐹   𝑦,𝐺   𝑦,𝐻   𝑆,𝑒,𝑛   𝑥,𝑆   𝑦,𝑆,𝑛   𝜑,𝑛
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑒)   𝐷(𝑦,𝑛)   𝐸(𝑦)   𝐹(𝑦,𝑛)   𝐺(𝑥,𝑒,𝑛)   𝐻(𝑥,𝑒,𝑛)   𝑉(𝑥,𝑦,𝑒,𝑛)

Proof of Theorem subsaliuncllem
StepHypRef Expression
1 subsaliuncllem.e . . 3 𝐸 = (𝐻𝐺)
2 subsaliuncllem.h . . . . . . 7 (𝜑𝐻 Fn ran 𝐺)
3 subsaliuncllem.f . . . . . . . 8 𝑦𝜑
4 vex 3484 . . . . . . . . . . . . . 14 𝑦 ∈ V
5 subsaliuncllem.g . . . . . . . . . . . . . . 15 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65elrnmpt 5969 . . . . . . . . . . . . . 14 (𝑦 ∈ V → (𝑦 ∈ ran 𝐺 ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
74, 6ax-mp 5 . . . . . . . . . . . . 13 (𝑦 ∈ ran 𝐺 ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
87biimpi 216 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
9 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
10 ssrab2 4080 . . . . . . . . . . . . . . . . 17 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆
1110a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆)
129, 11eqsstrd 4018 . . . . . . . . . . . . . . 15 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆)
1312a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆))
1413rexlimiv 3148 . . . . . . . . . . . . 13 (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆)
1514a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆))
168, 15mpd 15 . . . . . . . . . . 11 (𝑦 ∈ ran 𝐺𝑦𝑆)
1716adantl 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → 𝑦𝑆)
18 subsaliuncllem.y . . . . . . . . . . 11 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
1918r19.21bi 3251 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑦)
2017, 19sseldd 3984 . . . . . . . . 9 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑆)
2120ex 412 . . . . . . . 8 (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻𝑦) ∈ 𝑆))
223, 21ralrimi 3257 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆)
232, 22jca 511 . . . . . 6 (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
24 ffnfv 7139 . . . . . 6 (𝐻:ran 𝐺𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
2523, 24sylibr 234 . . . . 5 (𝜑𝐻:ran 𝐺𝑆)
26 eqid 2737 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
27 subsaliuncllem.s . . . . . . . . 9 (𝜑𝑆𝑉)
2826, 27rabexd 5340 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
2928ralrimivw 3150 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
305fnmpt 6708 . . . . . . 7 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → 𝐺 Fn ℕ)
3129, 30syl 17 . . . . . 6 (𝜑𝐺 Fn ℕ)
32 dffn3 6748 . . . . . 6 (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
3331, 32sylib 218 . . . . 5 (𝜑𝐺:ℕ⟶ran 𝐺)
34 fco 6760 . . . . 5 ((𝐻:ran 𝐺𝑆𝐺:ℕ⟶ran 𝐺) → (𝐻𝐺):ℕ⟶𝑆)
3525, 33, 34syl2anc 584 . . . 4 (𝜑 → (𝐻𝐺):ℕ⟶𝑆)
36 nnex 12272 . . . . . 6 ℕ ∈ V
3736a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
3827, 37elmapd 8880 . . . 4 (𝜑 → ((𝐻𝐺) ∈ (𝑆m ℕ) ↔ (𝐻𝐺):ℕ⟶𝑆))
3935, 38mpbird 257 . . 3 (𝜑 → (𝐻𝐺) ∈ (𝑆m ℕ))
401, 39eqeltrid 2845 . 2 (𝜑𝐸 ∈ (𝑆m ℕ))
4133ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ran 𝐺)
4218adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
43 fveq2 6906 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → (𝐻𝑦) = (𝐻‘(𝐺𝑛)))
44 id 22 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → 𝑦 = (𝐺𝑛))
4543, 44eleq12d 2835 . . . . . . . 8 (𝑦 = (𝐺𝑛) → ((𝐻𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
4645rspcva 3620 . . . . . . 7 (((𝐺𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4741, 42, 46syl2anc 584 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4833ffund 6740 . . . . . . . . 9 (𝜑 → Fun 𝐺)
4948adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Fun 𝐺)
50 simpr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
515dmeqi 5915 . . . . . . . . . . . . 13 dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
5251a1i 11 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
53 dmmptg 6262 . . . . . . . . . . . . 13 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = ℕ)
5429, 53syl 17 . . . . . . . . . . . 12 (𝜑 → dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = ℕ)
5552, 54eqtrd 2777 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = ℕ)
5655eqcomd 2743 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐺)
5756adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ℕ = dom 𝐺)
5850, 57eleqtrd 2843 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
5949, 58, 1fvcod 45232 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) = (𝐻‘(𝐺𝑛)))
605a1i 11 . . . . . . . . 9 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6128adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
6260, 61fvmpt2d 7029 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6362eqcomd 2743 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = (𝐺𝑛))
6459, 63eleq12d 2835 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
6547, 64mpbird 257 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
66 ineq1 4213 . . . . . . 7 (𝑥 = (𝐸𝑛) → (𝑥𝐷) = ((𝐸𝑛) ∩ 𝐷))
6766eqeq2d 2748 . . . . . 6 (𝑥 = (𝐸𝑛) → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6867elrab 3692 . . . . 5 ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6965, 68sylib 218 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7069simprd 495 . . 3 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
7170ralrimiva 3146 . 2 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
72 fveq1 6905 . . . . . 6 (𝑒 = 𝐸 → (𝑒𝑛) = (𝐸𝑛))
7372ineq1d 4219 . . . . 5 (𝑒 = 𝐸 → ((𝑒𝑛) ∩ 𝐷) = ((𝐸𝑛) ∩ 𝐷))
7473eqeq2d 2748 . . . 4 (𝑒 = 𝐸 → ((𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7574ralbidv 3178 . . 3 (𝑒 = 𝐸 → (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7675rspcev 3622 . 2 ((𝐸 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)) → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
7740, 71, 76syl2anc 584 1 (𝜑 → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wnf 1783  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  wss 3951  cmpt 5225  dom cdm 5685  ran crn 5686  ccom 5689  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cn 12266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-nn 12267
This theorem is referenced by:  subsaliuncl  46373
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