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Theorem subsaliuncllem 46313
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 3482 . . . . . . . . . . . . . 14 𝑦 ∈ V
5 subsaliuncllem.g . . . . . . . . . . . . . . 15 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65elrnmpt 5972 . . . . . . . . . . . . . 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 4090 . . . . . . . . . . . . . . . . 17 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆
1110a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆)
129, 11eqsstrd 4034 . . . . . . . . . . . . . . 15 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆)
1312a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆))
1413rexlimiv 3146 . . . . . . . . . . . . 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 3249 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑦)
2017, 19sseldd 3996 . . . . . . . . 9 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑆)
2120ex 412 . . . . . . . 8 (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻𝑦) ∈ 𝑆))
223, 21ralrimi 3255 . . . . . . 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 2735 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
27 subsaliuncllem.s . . . . . . . . 9 (𝜑𝑆𝑉)
2826, 27rabexd 5346 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
2928ralrimivw 3148 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
305fnmpt 6709 . . . . . . 7 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → 𝐺 Fn ℕ)
3129, 30syl 17 . . . . . 6 (𝜑𝐺 Fn ℕ)
32 dffn3 6749 . . . . . 6 (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
3331, 32sylib 218 . . . . 5 (𝜑𝐺:ℕ⟶ran 𝐺)
34 fco 6761 . . . . 5 ((𝐻:ran 𝐺𝑆𝐺:ℕ⟶ran 𝐺) → (𝐻𝐺):ℕ⟶𝑆)
3525, 33, 34syl2anc 584 . . . 4 (𝜑 → (𝐻𝐺):ℕ⟶𝑆)
36 nnex 12270 . . . . . 6 ℕ ∈ V
3736a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
3827, 37elmapd 8879 . . . 4 (𝜑 → ((𝐻𝐺) ∈ (𝑆m ℕ) ↔ (𝐻𝐺):ℕ⟶𝑆))
3935, 38mpbird 257 . . 3 (𝜑 → (𝐻𝐺) ∈ (𝑆m ℕ))
401, 39eqeltrid 2843 . 2 (𝜑𝐸 ∈ (𝑆m ℕ))
4133ffvelcdmda 7104 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ran 𝐺)
4218adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
43 fveq2 6907 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → (𝐻𝑦) = (𝐻‘(𝐺𝑛)))
44 id 22 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → 𝑦 = (𝐺𝑛))
4543, 44eleq12d 2833 . . . . . . . 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 6741 . . . . . . . . 9 (𝜑 → Fun 𝐺)
4948adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Fun 𝐺)
50 simpr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
515dmeqi 5918 . . . . . . . . . . . . 13 dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
5251a1i 11 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
53 dmmptg 6264 . . . . . . . . . . . . 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 2775 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = ℕ)
5655eqcomd 2741 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐺)
5756adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ℕ = dom 𝐺)
5850, 57eleqtrd 2841 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
5949, 58, 1fvcod 45170 . . . . . . 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 2741 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = (𝐺𝑛))
6459, 63eleq12d 2833 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
6547, 64mpbird 257 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
66 ineq1 4221 . . . . . . 7 (𝑥 = (𝐸𝑛) → (𝑥𝐷) = ((𝐸𝑛) ∩ 𝐷))
6766eqeq2d 2746 . . . . . 6 (𝑥 = (𝐸𝑛) → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6867elrab 3695 . . . . 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 3144 . 2 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
72 fveq1 6906 . . . . . 6 (𝑒 = 𝐸 → (𝑒𝑛) = (𝐸𝑛))
7372ineq1d 4227 . . . . 5 (𝑒 = 𝐸 → ((𝑒𝑛) ∩ 𝐷) = ((𝐸𝑛) ∩ 𝐷))
7473eqeq2d 2746 . . . 4 (𝑒 = 𝐸 → ((𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7574ralbidv 3176 . . 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 1537  wnf 1780  wcel 2106  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  cin 3962  wss 3963  cmpt 5231  dom cdm 5689  ran crn 5690  ccom 5693  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  cn 12264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867  df-nn 12265
This theorem is referenced by:  subsaliuncl  46314
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