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

Theorem subsaliuncl 45074
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
subsaliuncl.1 (𝜑𝑆 ∈ SAlg)
subsaliuncl.2 (𝜑𝐷𝑉)
subsaliuncl.3 𝑇 = (𝑆t 𝐷)
subsaliuncl.4 (𝜑𝐹:ℕ⟶𝑇)
Assertion
Ref Expression
subsaliuncl (𝜑 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)
Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑆,𝑛   𝜑,𝑛
Allowed substitution hints:   𝑇(𝑛)   𝑉(𝑛)

Proof of Theorem subsaliuncl
Dummy variables 𝑒 𝑓 𝑧 𝑚 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
2 subsaliuncl.1 . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
31, 2rabexd 5334 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
43ralrimivw 3151 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
5 eqid 2733 . . . . . . . 8 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65fnmpt 6691 . . . . . . 7 (∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V → (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) Fn ℕ)
74, 6syl 17 . . . . . 6 (𝜑 → (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) Fn ℕ)
8 nnex 12218 . . . . . . 7 ℕ ∈ V
9 fnrndomg 10531 . . . . . . 7 (ℕ ∈ V → ((𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) Fn ℕ → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ℕ))
108, 9ax-mp 5 . . . . . 6 ((𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) Fn ℕ → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ℕ)
117, 10syl 17 . . . . 5 (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ℕ)
12 nnenom 13945 . . . . . 6 ℕ ≈ ω
1312a1i 11 . . . . 5 (𝜑 → ℕ ≈ ω)
14 domentr 9009 . . . . 5 ((ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ℕ ∧ ℕ ≈ ω) → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ω)
1511, 13, 14syl2anc 585 . . . 4 (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ω)
16 vex 3479 . . . . . . . 8 𝑦 ∈ V
175elrnmpt 5956 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
1816, 17ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
1918biimpi 215 . . . . . 6 (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
2019adantl 483 . . . . 5 ((𝜑𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
21 simp3 1139 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
22 subsaliuncl.4 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶𝑇)
2322ffvelcdmda 7087 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ 𝑇)
24 subsaliuncl.3 . . . . . . . . . . . . 13 𝑇 = (𝑆t 𝐷)
2523, 24eleqtrdi 2844 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (𝑆t 𝐷))
26 subsaliuncl.2 . . . . . . . . . . . . . . 15 (𝜑𝐷𝑉)
2726elexd 3495 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ V)
28 elrest 17373 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
292, 27, 28syl2anc 585 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3029adantr 482 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3125, 30mpbid 231 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
32 rabn0 4386 . . . . . . . . . . 11 ({𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅ ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
3331, 32sylibr 233 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
34333adant3 1133 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
3521, 34eqnetrd 3009 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 ≠ ∅)
36353exp 1120 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅)))
3736rexlimdv 3154 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅))
3837adantr 482 . . . . 5 ((𝜑𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})) → (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅))
3920, 38mpd 15 . . . 4 ((𝜑𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})) → 𝑦 ≠ ∅)
4015, 39axccdom 43921 . . 3 (𝜑 → ∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦))
41 simpl 484 . . . . . 6 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → 𝜑)
42 fveq2 6892 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
4342eqeq1d 2735 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑥𝐷)))
4443rabbidv 3441 . . . . . . . . . . 11 (𝑛 = 𝑚 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4544cbvmptv 5262 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4645rneqi 5937 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4746fneq2i 6648 . . . . . . . 8 (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ↔ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
4847biimpi 215 . . . . . . 7 (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
4948ad2antrl 727 . . . . . 6 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
5046raleqi 3324 . . . . . . . . 9 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5150biimpi 215 . . . . . . . 8 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5251adantl 483 . . . . . . 7 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5352adantrl 715 . . . . . 6 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
54 nfv 1918 . . . . . . 7 𝑧(𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5523ad2ant1 1134 . . . . . . 7 ((𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
56 ineq1 4206 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐷) = (𝑧𝐷))
5756eqeq2d 2744 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑧𝐷)))
5857cbvrabv 3443 . . . . . . . . . 10 {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)} = {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}
5958mpteq2i 5254 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})
6045, 59eqtr2i 2762 . . . . . . . 8 (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6160coeq2i 5861 . . . . . . 7 (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6247biimpri 227 . . . . . . . 8 (𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) → 𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
63623ad2ant2 1135 . . . . . . 7 ((𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → 𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6446eqcomi 2742 . . . . . . . . . . 11 ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6564raleqi 3324 . . . . . . . . . 10 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
66 fveq2 6892 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
67 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑦 = 𝑧)
6866, 67eleq12d 2828 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑧) ∈ 𝑧))
6968cbvralvw 3235 . . . . . . . . . 10 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
7065, 69bitri 275 . . . . . . . . 9 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
7170biimpi 215 . . . . . . . 8 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
72713ad2ant3 1136 . . . . . . 7 ((𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
7354, 55, 5, 61, 63, 72subsaliuncllem 45073 . . . . . 6 ((𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
7441, 49, 53, 73syl3anc 1372 . . . . 5 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
7574ex 414 . . . 4 (𝜑 → ((𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)))
7675exlimdv 1937 . . 3 (𝜑 → (∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)))
7740, 76mpd 15 . 2 (𝜑 → ∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
7823ad2ant1 1134 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
79273ad2ant1 1134 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝐷 ∈ V)
802adantr 482 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑆 ∈ SAlg)
81 nnct 13946 . . . . . . . . 9 ℕ ≼ ω
8281a1i 11 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆m ℕ)) → ℕ ≼ ω)
83 elmapi 8843 . . . . . . . . . 10 (𝑒 ∈ (𝑆m ℕ) → 𝑒:ℕ⟶𝑆)
8483adantl 483 . . . . . . . . 9 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑒:ℕ⟶𝑆)
8584ffvelcdmda 7087 . . . . . . . 8 (((𝜑𝑒 ∈ (𝑆m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒𝑛) ∈ 𝑆)
8680, 82, 85saliuncl 45039 . . . . . . 7 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)
87863adant3 1133 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)
88 eqid 2733 . . . . . 6 ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷)
8978, 79, 87, 88elrestd 43797 . . . . 5 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷))
90 nfra1 3282 . . . . . . . . 9 𝑛𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)
91 rspa 3246 . . . . . . . . 9 ((∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
9290, 91iuneq2df 43733 . . . . . . . 8 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = 𝑛 ∈ ℕ ((𝑒𝑛) ∩ 𝐷))
93 iunin1 5076 . . . . . . . . 9 𝑛 ∈ ℕ ((𝑒𝑛) ∩ 𝐷) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷)
9493a1i 11 . . . . . . . 8 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ ((𝑒𝑛) ∩ 𝐷) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷))
9592, 94eqtrd 2773 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷))
96953ad2ant3 1136 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝐹𝑛) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷))
9724a1i 11 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑇 = (𝑆t 𝐷))
9896, 97eleq12d 2828 . . . . 5 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇 ↔ ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷)))
9989, 98mpbird 257 . . . 4 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)
100993exp 1120 . . 3 (𝜑 → (𝑒 ∈ (𝑆m ℕ) → (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)))
101100rexlimdv 3154 . 2 (𝜑 → (∃𝑒 ∈ (𝑆m ℕ)∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇))
10277, 101mpd 15 1 (𝜑 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cin 3948  c0 4323   ciun 4998   class class class wbr 5149  cmpt 5232  ran crn 5678  ccom 5681   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  ωcom 7855  m cmap 8820  cen 8936  cdom 8937  cn 12212  t crest 17366  SAlgcsalg 45024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-cc 10430  ax-ac2 10458  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-acn 9937  df-ac 10111  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-rest 17368  df-salg 45025
This theorem is referenced by:  subsalsal  45075
  Copyright terms: Public domain W3C validator