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Theorem subsaliuncllem 45372
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 β„²π‘¦πœ‘
subsaliuncllem.s (πœ‘ β†’ 𝑆 ∈ 𝑉)
subsaliuncllem.g 𝐺 = (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
subsaliuncllem.e 𝐸 = (𝐻 ∘ 𝐺)
subsaliuncllem.h (πœ‘ β†’ 𝐻 Fn ran 𝐺)
subsaliuncllem.y (πœ‘ β†’ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑦)
Assertion
Ref Expression
subsaliuncllem (πœ‘ β†’ βˆƒπ‘’ ∈ (𝑆 ↑m β„•)βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((π‘’β€˜π‘›) ∩ 𝐷))
Distinct variable groups:   𝐷,𝑒   π‘₯,𝐷   𝑒,𝐸,𝑛   π‘₯,𝐸,𝑛   𝑒,𝐹   π‘₯,𝐹   𝑦,𝐺   𝑦,𝐻   𝑆,𝑒,𝑛   π‘₯,𝑆   𝑦,𝑆,𝑛   πœ‘,𝑛
Allowed substitution hints:   πœ‘(π‘₯,𝑦,𝑒)   𝐷(𝑦,𝑛)   𝐸(𝑦)   𝐹(𝑦,𝑛)   𝐺(π‘₯,𝑒,𝑛)   𝐻(π‘₯,𝑒,𝑛)   𝑉(π‘₯,𝑦,𝑒,𝑛)

Proof of Theorem subsaliuncllem
StepHypRef Expression
1 subsaliuncllem.e . . 3 𝐸 = (𝐻 ∘ 𝐺)
2 subsaliuncllem.h . . . . . . 7 (πœ‘ β†’ 𝐻 Fn ran 𝐺)
3 subsaliuncllem.f . . . . . . . 8 β„²π‘¦πœ‘
4 vex 3477 . . . . . . . . . . . . . 14 𝑦 ∈ V
5 subsaliuncllem.g . . . . . . . . . . . . . . 15 𝐺 = (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
65elrnmpt 5955 . . . . . . . . . . . . . 14 (𝑦 ∈ V β†’ (𝑦 ∈ ran 𝐺 ↔ βˆƒπ‘› ∈ β„• 𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)}))
74, 6ax-mp 5 . . . . . . . . . . . . 13 (𝑦 ∈ ran 𝐺 ↔ βˆƒπ‘› ∈ β„• 𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
87biimpi 215 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 β†’ βˆƒπ‘› ∈ β„• 𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
9 id 22 . . . . . . . . . . . . . . . 16 (𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} β†’ 𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
10 ssrab2 4077 . . . . . . . . . . . . . . . . 17 {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} βŠ† 𝑆
1110a1i 11 . . . . . . . . . . . . . . . 16 (𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} β†’ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} βŠ† 𝑆)
129, 11eqsstrd 4020 . . . . . . . . . . . . . . 15 (𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} β†’ 𝑦 βŠ† 𝑆)
1312a1i 11 . . . . . . . . . . . . . 14 (𝑛 ∈ β„• β†’ (𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} β†’ 𝑦 βŠ† 𝑆))
1413rexlimiv 3147 . . . . . . . . . . . . 13 (βˆƒπ‘› ∈ β„• 𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} β†’ 𝑦 βŠ† 𝑆)
1514a1i 11 . . . . . . . . . . . 12 (𝑦 ∈ ran 𝐺 β†’ (βˆƒπ‘› ∈ β„• 𝑦 = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} β†’ 𝑦 βŠ† 𝑆))
168, 15mpd 15 . . . . . . . . . . 11 (𝑦 ∈ ran 𝐺 β†’ 𝑦 βŠ† 𝑆)
1716adantl 481 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ ran 𝐺) β†’ 𝑦 βŠ† 𝑆)
18 subsaliuncllem.y . . . . . . . . . . 11 (πœ‘ β†’ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑦)
1918r19.21bi 3247 . . . . . . . . . 10 ((πœ‘ ∧ 𝑦 ∈ ran 𝐺) β†’ (π»β€˜π‘¦) ∈ 𝑦)
2017, 19sseldd 3983 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ ran 𝐺) β†’ (π»β€˜π‘¦) ∈ 𝑆)
2120ex 412 . . . . . . . 8 (πœ‘ β†’ (𝑦 ∈ ran 𝐺 β†’ (π»β€˜π‘¦) ∈ 𝑆))
223, 21ralrimi 3253 . . . . . . 7 (πœ‘ β†’ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑆)
232, 22jca 511 . . . . . 6 (πœ‘ β†’ (𝐻 Fn ran 𝐺 ∧ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑆))
24 ffnfv 7120 . . . . . 6 (𝐻:ran πΊβŸΆπ‘† ↔ (𝐻 Fn ran 𝐺 ∧ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑆))
2523, 24sylibr 233 . . . . 5 (πœ‘ β†’ 𝐻:ran πΊβŸΆπ‘†)
26 eqid 2731 . . . . . . . . 9 {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)}
27 subsaliuncllem.s . . . . . . . . 9 (πœ‘ β†’ 𝑆 ∈ 𝑉)
2826, 27rabexd 5333 . . . . . . . 8 (πœ‘ β†’ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ∈ V)
2928ralrimivw 3149 . . . . . . 7 (πœ‘ β†’ βˆ€π‘› ∈ β„• {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ∈ V)
305fnmpt 6690 . . . . . . 7 (βˆ€π‘› ∈ β„• {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ∈ V β†’ 𝐺 Fn β„•)
3129, 30syl 17 . . . . . 6 (πœ‘ β†’ 𝐺 Fn β„•)
32 dffn3 6730 . . . . . 6 (𝐺 Fn β„• ↔ 𝐺:β„•βŸΆran 𝐺)
3331, 32sylib 217 . . . . 5 (πœ‘ β†’ 𝐺:β„•βŸΆran 𝐺)
34 fco 6741 . . . . 5 ((𝐻:ran πΊβŸΆπ‘† ∧ 𝐺:β„•βŸΆran 𝐺) β†’ (𝐻 ∘ 𝐺):β„•βŸΆπ‘†)
3525, 33, 34syl2anc 583 . . . 4 (πœ‘ β†’ (𝐻 ∘ 𝐺):β„•βŸΆπ‘†)
36 nnex 12223 . . . . . 6 β„• ∈ V
3736a1i 11 . . . . 5 (πœ‘ β†’ β„• ∈ V)
3827, 37elmapd 8838 . . . 4 (πœ‘ β†’ ((𝐻 ∘ 𝐺) ∈ (𝑆 ↑m β„•) ↔ (𝐻 ∘ 𝐺):β„•βŸΆπ‘†))
3935, 38mpbird 257 . . 3 (πœ‘ β†’ (𝐻 ∘ 𝐺) ∈ (𝑆 ↑m β„•))
401, 39eqeltrid 2836 . 2 (πœ‘ β†’ 𝐸 ∈ (𝑆 ↑m β„•))
4133ffvelcdmda 7086 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) ∈ ran 𝐺)
4218adantr 480 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑦)
43 fveq2 6891 . . . . . . . . 9 (𝑦 = (πΊβ€˜π‘›) β†’ (π»β€˜π‘¦) = (π»β€˜(πΊβ€˜π‘›)))
44 id 22 . . . . . . . . 9 (𝑦 = (πΊβ€˜π‘›) β†’ 𝑦 = (πΊβ€˜π‘›))
4543, 44eleq12d 2826 . . . . . . . 8 (𝑦 = (πΊβ€˜π‘›) β†’ ((π»β€˜π‘¦) ∈ 𝑦 ↔ (π»β€˜(πΊβ€˜π‘›)) ∈ (πΊβ€˜π‘›)))
4645rspcva 3610 . . . . . . 7 (((πΊβ€˜π‘›) ∈ ran 𝐺 ∧ βˆ€π‘¦ ∈ ran 𝐺(π»β€˜π‘¦) ∈ 𝑦) β†’ (π»β€˜(πΊβ€˜π‘›)) ∈ (πΊβ€˜π‘›))
4741, 42, 46syl2anc 583 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (π»β€˜(πΊβ€˜π‘›)) ∈ (πΊβ€˜π‘›))
4833ffund 6721 . . . . . . . . 9 (πœ‘ β†’ Fun 𝐺)
4948adantr 480 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ Fun 𝐺)
50 simpr 484 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ β„•)
515dmeqi 5904 . . . . . . . . . . . . 13 dom 𝐺 = dom (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
5251a1i 11 . . . . . . . . . . . 12 (πœ‘ β†’ dom 𝐺 = dom (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)}))
53 dmmptg 6241 . . . . . . . . . . . . 13 (βˆ€π‘› ∈ β„• {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ∈ V β†’ dom (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)}) = β„•)
5429, 53syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ dom (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)}) = β„•)
5552, 54eqtrd 2771 . . . . . . . . . . 11 (πœ‘ β†’ dom 𝐺 = β„•)
5655eqcomd 2737 . . . . . . . . . 10 (πœ‘ β†’ β„• = dom 𝐺)
5756adantr 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ β„• = dom 𝐺)
5850, 57eleqtrd 2834 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ 𝑛 ∈ dom 𝐺)
5949, 58, 1fvcod 44225 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΈβ€˜π‘›) = (π»β€˜(πΊβ€˜π‘›)))
605a1i 11 . . . . . . . . 9 (πœ‘ β†’ 𝐺 = (𝑛 ∈ β„• ↦ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)}))
6128adantr 480 . . . . . . . . 9 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ∈ V)
6260, 61fvmpt2d 7011 . . . . . . . 8 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΊβ€˜π‘›) = {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
6362eqcomd 2737 . . . . . . 7 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} = (πΊβ€˜π‘›))
6459, 63eleq12d 2826 . . . . . 6 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΈβ€˜π‘›) ∈ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ↔ (π»β€˜(πΊβ€˜π‘›)) ∈ (πΊβ€˜π‘›)))
6547, 64mpbird 257 . . . . 5 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΈβ€˜π‘›) ∈ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)})
66 ineq1 4205 . . . . . . 7 (π‘₯ = (πΈβ€˜π‘›) β†’ (π‘₯ ∩ 𝐷) = ((πΈβ€˜π‘›) ∩ 𝐷))
6766eqeq2d 2742 . . . . . 6 (π‘₯ = (πΈβ€˜π‘›) β†’ ((πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷) ↔ (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷)))
6867elrab 3683 . . . . 5 ((πΈβ€˜π‘›) ∈ {π‘₯ ∈ 𝑆 ∣ (πΉβ€˜π‘›) = (π‘₯ ∩ 𝐷)} ↔ ((πΈβ€˜π‘›) ∈ 𝑆 ∧ (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷)))
6965, 68sylib 217 . . . 4 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ ((πΈβ€˜π‘›) ∈ 𝑆 ∧ (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷)))
7069simprd 495 . . 3 ((πœ‘ ∧ 𝑛 ∈ β„•) β†’ (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷))
7170ralrimiva 3145 . 2 (πœ‘ β†’ βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷))
72 fveq1 6890 . . . . . 6 (𝑒 = 𝐸 β†’ (π‘’β€˜π‘›) = (πΈβ€˜π‘›))
7372ineq1d 4211 . . . . 5 (𝑒 = 𝐸 β†’ ((π‘’β€˜π‘›) ∩ 𝐷) = ((πΈβ€˜π‘›) ∩ 𝐷))
7473eqeq2d 2742 . . . 4 (𝑒 = 𝐸 β†’ ((πΉβ€˜π‘›) = ((π‘’β€˜π‘›) ∩ 𝐷) ↔ (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷)))
7574ralbidv 3176 . . 3 (𝑒 = 𝐸 β†’ (βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((π‘’β€˜π‘›) ∩ 𝐷) ↔ βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷)))
7675rspcev 3612 . 2 ((𝐸 ∈ (𝑆 ↑m β„•) ∧ βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((πΈβ€˜π‘›) ∩ 𝐷)) β†’ βˆƒπ‘’ ∈ (𝑆 ↑m β„•)βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((π‘’β€˜π‘›) ∩ 𝐷))
7740, 71, 76syl2anc 583 1 (πœ‘ β†’ βˆƒπ‘’ ∈ (𝑆 ↑m β„•)βˆ€π‘› ∈ β„• (πΉβ€˜π‘›) = ((π‘’β€˜π‘›) ∩ 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540  β„²wnf 1784   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431  Vcvv 3473   ∩ cin 3947   βŠ† wss 3948   ↦ cmpt 5231  dom cdm 5676  ran crn 5677   ∘ ccom 5680  Fun wfun 6537   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412   ↑m cmap 8824  β„•cn 12217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-1cn 11172  ax-addcl 11174
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-map 8826  df-nn 12218
This theorem is referenced by:  subsaliuncl  45373
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