Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  subsaliuncllem Structured version   Visualization version   GIF version

Theorem subsaliuncllem 39251
 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 3176 . . . . . . . . . . . . . 14 𝑦 ∈ V
5 subsaliuncllem.g . . . . . . . . . . . . . . 15 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65elrnmpt 5293 . . . . . . . . . . . . . 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 205 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
9 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
10 ssrab2 3650 . . . . . . . . . . . . . . . . 17 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆
1110a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆)
129, 11eqsstrd 3602 . . . . . . . . . . . . . . 15 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆)
1312a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆))
1413rexlimiv 3009 . . . . . . . . . . . . 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 2916 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑦)
2017, 19sseldd 3569 . . . . . . . . 9 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑆)
2120ex 449 . . . . . . . 8 (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻𝑦) ∈ 𝑆))
223, 21ralrimi 2940 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆)
232, 22jca 553 . . . . . 6 (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
24 ffnfv 6295 . . . . . 6 (𝐻:ran 𝐺𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
2523, 24sylibr 223 . . . . 5 (𝜑𝐻:ran 𝐺𝑆)
26 eqid 2610 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
27 subsaliuncllem.s . . . . . . . . 9 (𝜑𝑆𝑉)
2826, 27rabexd 4741 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
2928ralrimivw 2950 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
305fnmpt 5933 . . . . . . 7 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → 𝐺 Fn ℕ)
3129, 30syl 17 . . . . . 6 (𝜑𝐺 Fn ℕ)
32 dffn3 5967 . . . . . 6 (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
3331, 32sylib 207 . . . . 5 (𝜑𝐺:ℕ⟶ran 𝐺)
34 fco 5971 . . . . 5 ((𝐻:ran 𝐺𝑆𝐺:ℕ⟶ran 𝐺) → (𝐻𝐺):ℕ⟶𝑆)
3525, 33, 34syl2anc 691 . . . 4 (𝜑 → (𝐻𝐺):ℕ⟶𝑆)
36 nnex 10903 . . . . . 6 ℕ ∈ V
3736a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
3827, 37elmapd 7758 . . . 4 (𝜑 → ((𝐻𝐺) ∈ (𝑆𝑚 ℕ) ↔ (𝐻𝐺):ℕ⟶𝑆))
3935, 38mpbird 246 . . 3 (𝜑 → (𝐻𝐺) ∈ (𝑆𝑚 ℕ))
401, 39syl5eqel 2692 . 2 (𝜑𝐸 ∈ (𝑆𝑚 ℕ))
4133ffvelrnda 6267 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ran 𝐺)
4218adantr 480 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
43 fveq2 6103 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → (𝐻𝑦) = (𝐻‘(𝐺𝑛)))
44 id 22 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → 𝑦 = (𝐺𝑛))
4543, 44eleq12d 2682 . . . . . . . 8 (𝑦 = (𝐺𝑛) → ((𝐻𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
4645rspcva 3280 . . . . . . 7 (((𝐺𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4741, 42, 46syl2anc 691 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4833ffund 5962 . . . . . . . . 9 (𝜑 → Fun 𝐺)
4948adantr 480 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Fun 𝐺)
50 simpr 476 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
515dmeqi 5247 . . . . . . . . . . . . 13 dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
5251a1i 11 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
53 dmmptg 5549 . . . . . . . . . . . . 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 2644 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = ℕ)
5655eqcomd 2616 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐺)
5756adantr 480 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ℕ = dom 𝐺)
5850, 57eleqtrd 2690 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
5949, 58, 1fvcod 38418 . . . . . . 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 6202 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6362eqcomd 2616 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = (𝐺𝑛))
6459, 63eleq12d 2682 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
6547, 64mpbird 246 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
66 ineq1 3769 . . . . . . 7 (𝑥 = (𝐸𝑛) → (𝑥𝐷) = ((𝐸𝑛) ∩ 𝐷))
6766eqeq2d 2620 . . . . . 6 (𝑥 = (𝐸𝑛) → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6867elrab 3331 . . . . 5 ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6965, 68sylib 207 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7069simprd 478 . . 3 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
7170ralrimiva 2949 . 2 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
72 fveq1 6102 . . . . . 6 (𝑒 = 𝐸 → (𝑒𝑛) = (𝐸𝑛))
7372ineq1d 3775 . . . . 5 (𝑒 = 𝐸 → ((𝑒𝑛) ∩ 𝐷) = ((𝐸𝑛) ∩ 𝐷))
7473eqeq2d 2620 . . . 4 (𝑒 = 𝐸 → ((𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7574ralbidv 2969 . . 3 (𝑒 = 𝐸 → (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7675rspcev 3282 . 2 ((𝐸 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)) → ∃𝑒 ∈ (𝑆𝑚 ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
7740, 71, 76syl2anc 691 1 (𝜑 → ∃𝑒 ∈ (𝑆𝑚 ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ∘ ccom 5042  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℕcn 10897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rrecex 9887  ax-cnre 9888 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-map 7746  df-nn 10898 This theorem is referenced by:  subsaliuncl  39252
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