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Theorem subsaliuncllem 44588
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 3449 . . . . . . . . . . . . . 14 𝑦 ∈ V
5 subsaliuncllem.g . . . . . . . . . . . . . . 15 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65elrnmpt 5911 . . . . . . . . . . . . . 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 215 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
9 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
10 ssrab2 4037 . . . . . . . . . . . . . . . . 17 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆
1110a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆)
129, 11eqsstrd 3982 . . . . . . . . . . . . . . 15 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆)
1312a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆))
1413rexlimiv 3145 . . . . . . . . . . . . 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 482 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → 𝑦𝑆)
18 subsaliuncllem.y . . . . . . . . . . 11 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
1918r19.21bi 3234 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑦)
2017, 19sseldd 3945 . . . . . . . . 9 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑆)
2120ex 413 . . . . . . . 8 (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻𝑦) ∈ 𝑆))
223, 21ralrimi 3240 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆)
232, 22jca 512 . . . . . 6 (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
24 ffnfv 7066 . . . . . 6 (𝐻:ran 𝐺𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
2523, 24sylibr 233 . . . . 5 (𝜑𝐻:ran 𝐺𝑆)
26 eqid 2736 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
27 subsaliuncllem.s . . . . . . . . 9 (𝜑𝑆𝑉)
2826, 27rabexd 5290 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
2928ralrimivw 3147 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
305fnmpt 6641 . . . . . . 7 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → 𝐺 Fn ℕ)
3129, 30syl 17 . . . . . 6 (𝜑𝐺 Fn ℕ)
32 dffn3 6681 . . . . . 6 (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
3331, 32sylib 217 . . . . 5 (𝜑𝐺:ℕ⟶ran 𝐺)
34 fco 6692 . . . . 5 ((𝐻:ran 𝐺𝑆𝐺:ℕ⟶ran 𝐺) → (𝐻𝐺):ℕ⟶𝑆)
3525, 33, 34syl2anc 584 . . . 4 (𝜑 → (𝐻𝐺):ℕ⟶𝑆)
36 nnex 12159 . . . . . 6 ℕ ∈ V
3736a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
3827, 37elmapd 8779 . . . 4 (𝜑 → ((𝐻𝐺) ∈ (𝑆m ℕ) ↔ (𝐻𝐺):ℕ⟶𝑆))
3935, 38mpbird 256 . . 3 (𝜑 → (𝐻𝐺) ∈ (𝑆m ℕ))
401, 39eqeltrid 2842 . 2 (𝜑𝐸 ∈ (𝑆m ℕ))
4133ffvelcdmda 7035 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ran 𝐺)
4218adantr 481 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
43 fveq2 6842 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → (𝐻𝑦) = (𝐻‘(𝐺𝑛)))
44 id 22 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → 𝑦 = (𝐺𝑛))
4543, 44eleq12d 2832 . . . . . . . 8 (𝑦 = (𝐺𝑛) → ((𝐻𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
4645rspcva 3579 . . . . . . 7 (((𝐺𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4741, 42, 46syl2anc 584 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4833ffund 6672 . . . . . . . . 9 (𝜑 → Fun 𝐺)
4948adantr 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Fun 𝐺)
50 simpr 485 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
515dmeqi 5860 . . . . . . . . . . . . 13 dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
5251a1i 11 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
53 dmmptg 6194 . . . . . . . . . . . . 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 2776 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = ℕ)
5655eqcomd 2742 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐺)
5756adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ℕ = dom 𝐺)
5850, 57eleqtrd 2840 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
5949, 58, 1fvcod 43438 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) = (𝐻‘(𝐺𝑛)))
605a1i 11 . . . . . . . . 9 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6128adantr 481 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
6260, 61fvmpt2d 6961 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6362eqcomd 2742 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = (𝐺𝑛))
6459, 63eleq12d 2832 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
6547, 64mpbird 256 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
66 ineq1 4165 . . . . . . 7 (𝑥 = (𝐸𝑛) → (𝑥𝐷) = ((𝐸𝑛) ∩ 𝐷))
6766eqeq2d 2747 . . . . . 6 (𝑥 = (𝐸𝑛) → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6867elrab 3645 . . . . 5 ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6965, 68sylib 217 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7069simprd 496 . . 3 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
7170ralrimiva 3143 . 2 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
72 fveq1 6841 . . . . . 6 (𝑒 = 𝐸 → (𝑒𝑛) = (𝐸𝑛))
7372ineq1d 4171 . . . . 5 (𝑒 = 𝐸 → ((𝑒𝑛) ∩ 𝐷) = ((𝐸𝑛) ∩ 𝐷))
7473eqeq2d 2747 . . . 4 (𝑒 = 𝐸 → ((𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7574ralbidv 3174 . . 3 (𝑒 = 𝐸 → (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7675rspcev 3581 . 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 205  wa 396   = wceq 1541  wnf 1785  wcel 2106  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cin 3909  wss 3910  cmpt 5188  dom cdm 5633  ran crn 5634  ccom 5637  Fun wfun 6490   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  cn 12153
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-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111
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-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-iun 4956  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-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-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-map 8767  df-nn 12154
This theorem is referenced by:  subsaliuncl  44589
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