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

Theorem subsaliuncllem 43040
 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 3444 . . . . . . . . . . . . . 14 𝑦 ∈ V
5 subsaliuncllem.g . . . . . . . . . . . . . . 15 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65elrnmpt 5793 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
9 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
10 ssrab2 4007 . . . . . . . . . . . . . . . . 17 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆
1110a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ⊆ 𝑆)
129, 11eqsstrd 3953 . . . . . . . . . . . . . . 15 (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆)
1312a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦𝑆))
1413rexlimiv 3239 . . . . . . . . . . . . 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 485 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → 𝑦𝑆)
18 subsaliuncllem.y . . . . . . . . . . 11 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
1918r19.21bi 3173 . . . . . . . . . 10 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑦)
2017, 19sseldd 3916 . . . . . . . . 9 ((𝜑𝑦 ∈ ran 𝐺) → (𝐻𝑦) ∈ 𝑆)
2120ex 416 . . . . . . . 8 (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻𝑦) ∈ 𝑆))
223, 21ralrimi 3180 . . . . . . 7 (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆)
232, 22jca 515 . . . . . 6 (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
24 ffnfv 6860 . . . . . 6 (𝐻:ran 𝐺𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑆))
2523, 24sylibr 237 . . . . 5 (𝜑𝐻:ran 𝐺𝑆)
26 eqid 2798 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
27 subsaliuncllem.s . . . . . . . . 9 (𝜑𝑆𝑉)
2826, 27rabexd 5201 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
2928ralrimivw 3150 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
305fnmpt 6461 . . . . . . 7 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → 𝐺 Fn ℕ)
3129, 30syl 17 . . . . . 6 (𝜑𝐺 Fn ℕ)
32 dffn3 6500 . . . . . 6 (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
3331, 32sylib 221 . . . . 5 (𝜑𝐺:ℕ⟶ran 𝐺)
34 fco 6506 . . . . 5 ((𝐻:ran 𝐺𝑆𝐺:ℕ⟶ran 𝐺) → (𝐻𝐺):ℕ⟶𝑆)
3525, 33, 34syl2anc 587 . . . 4 (𝜑 → (𝐻𝐺):ℕ⟶𝑆)
36 nnex 11634 . . . . . 6 ℕ ∈ V
3736a1i 11 . . . . 5 (𝜑 → ℕ ∈ V)
3827, 37elmapd 8406 . . . 4 (𝜑 → ((𝐻𝐺) ∈ (𝑆m ℕ) ↔ (𝐻𝐺):ℕ⟶𝑆))
3935, 38mpbird 260 . . 3 (𝜑 → (𝐻𝐺) ∈ (𝑆m ℕ))
401, 39eqeltrid 2894 . 2 (𝜑𝐸 ∈ (𝑆m ℕ))
4133ffvelrnda 6829 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ ran 𝐺)
4218adantr 484 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦)
43 fveq2 6646 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → (𝐻𝑦) = (𝐻‘(𝐺𝑛)))
44 id 22 . . . . . . . . 9 (𝑦 = (𝐺𝑛) → 𝑦 = (𝐺𝑛))
4543, 44eleq12d 2884 . . . . . . . 8 (𝑦 = (𝐺𝑛) → ((𝐻𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
4645rspcva 3569 . . . . . . 7 (((𝐺𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻𝑦) ∈ 𝑦) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4741, 42, 46syl2anc 587 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛))
4833ffund 6492 . . . . . . . . 9 (𝜑 → Fun 𝐺)
4948adantr 484 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → Fun 𝐺)
50 simpr 488 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
515dmeqi 5738 . . . . . . . . . . . . 13 dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
5251a1i 11 . . . . . . . . . . . 12 (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
53 dmmptg 6064 . . . . . . . . . . . . 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 2833 . . . . . . . . . . 11 (𝜑 → dom 𝐺 = ℕ)
5655eqcomd 2804 . . . . . . . . . 10 (𝜑 → ℕ = dom 𝐺)
5756adantr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ℕ = dom 𝐺)
5850, 57eleqtrd 2892 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
5949, 58, 1fvcod 41901 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) = (𝐻‘(𝐺𝑛)))
605a1i 11 . . . . . . . . 9 (𝜑𝐺 = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6128adantr 484 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
6260, 61fvmpt2d 6759 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6362eqcomd 2804 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = (𝐺𝑛))
6459, 63eleq12d 2884 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ (𝐻‘(𝐺𝑛)) ∈ (𝐺𝑛)))
6547, 64mpbird 260 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
66 ineq1 4131 . . . . . . 7 (𝑥 = (𝐸𝑛) → (𝑥𝐷) = ((𝐸𝑛) ∩ 𝐷))
6766eqeq2d 2809 . . . . . 6 (𝑥 = (𝐸𝑛) → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6867elrab 3628 . . . . 5 ((𝐸𝑛) ∈ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ↔ ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
6965, 68sylib 221 . . . 4 ((𝜑𝑛 ∈ ℕ) → ((𝐸𝑛) ∈ 𝑆 ∧ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7069simprd 499 . . 3 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
7170ralrimiva 3149 . 2 (𝜑 → ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷))
72 fveq1 6645 . . . . . 6 (𝑒 = 𝐸 → (𝑒𝑛) = (𝐸𝑛))
7372ineq1d 4138 . . . . 5 (𝑒 = 𝐸 → ((𝑒𝑛) ∩ 𝐷) = ((𝐸𝑛) ∩ 𝐷))
7473eqeq2d 2809 . . . 4 (𝑒 = 𝐸 → ((𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7574ralbidv 3162 . . 3 (𝑒 = 𝐸 → (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ↔ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)))
7675rspcev 3571 . 2 ((𝐸 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝐸𝑛) ∩ 𝐷)) → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
7740, 71, 76syl2anc 587 1 (𝜑 → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881   ↦ cmpt 5111  dom cdm 5520  ran crn 5521   ∘ ccom 5524  Fun wfun 6319   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136   ↑m cmap 8392  ℕcn 11628 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-1cn 10587  ax-addcl 10589 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-map 8394  df-nn 11629 This theorem is referenced by:  subsaliuncl  43041
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