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Theorem subsaliuncl 46373
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 2737 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
2 subsaliuncl.1 . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
31, 2rabexd 5340 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
43ralrimivw 3150 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
5 eqid 2737 . . . . . . . 8 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65fnmpt 6708 . . . . . . 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 12272 . . . . . . 7 ℕ ∈ V
9 fnrndomg 10576 . . . . . . 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 14021 . . . . . 6 ℕ ≈ ω
1312a1i 11 . . . . 5 (𝜑 → ℕ ≈ ω)
14 domentr 9053 . . . . 5 ((ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ℕ ∧ ℕ ≈ ω) → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ω)
1511, 13, 14syl2anc 584 . . . 4 (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ω)
16 vex 3484 . . . . . . . 8 𝑦 ∈ V
175elrnmpt 5969 . . . . . . . 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 216 . . . . . 6 (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
2019adantl 481 . . . . 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 7104 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ 𝑇)
24 subsaliuncl.3 . . . . . . . . . . . . 13 𝑇 = (𝑆t 𝐷)
2523, 24eleqtrdi 2851 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (𝑆t 𝐷))
26 subsaliuncl.2 . . . . . . . . . . . . . . 15 (𝜑𝐷𝑉)
2726elexd 3504 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ V)
28 elrest 17472 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
292, 27, 28syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3029adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3125, 30mpbid 232 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
32 rabn0 4389 . . . . . . . . . . 11 ({𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅ ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
3331, 32sylibr 234 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
34333adant3 1133 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
3521, 34eqnetrd 3008 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 ≠ ∅)
36353exp 1120 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅)))
3736rexlimdv 3153 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅))
3837adantr 480 . . . . 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 45227 . . 3 (𝜑 → ∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦))
41 simpl 482 . . . . . 6 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → 𝜑)
42 fveq2 6906 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
4342eqeq1d 2739 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑥𝐷)))
4443rabbidv 3444 . . . . . . . . . . 11 (𝑛 = 𝑚 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4544cbvmptv 5255 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4645rneqi 5948 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4746fneq2i 6666 . . . . . . . 8 (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ↔ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
4847biimpi 216 . . . . . . 7 (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
4948ad2antrl 728 . . . . . 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 216 . . . . . . . 8 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5251adantl 481 . . . . . . 7 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5352adantrl 716 . . . . . 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 1914 . . . . . . 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 4213 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐷) = (𝑧𝐷))
5756eqeq2d 2748 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑧𝐷)))
5857cbvrabv 3447 . . . . . . . . . 10 {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)} = {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}
5958mpteq2i 5247 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})
6045, 59eqtr2i 2766 . . . . . . . 8 (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6160coeq2i 5871 . . . . . . 7 (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6247biimpri 228 . . . . . . . 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 2746 . . . . . . . . . . 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 6906 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
67 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑦 = 𝑧)
6866, 67eleq12d 2835 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑧) ∈ 𝑧))
6968cbvralvw 3237 . . . . . . . . . 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 216 . . . . . . . 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 46372 . . . . . 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 1373 . . . . 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 412 . . . 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 1933 . . 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 480 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑆 ∈ SAlg)
81 nnct 14022 . . . . . . . . 9 ℕ ≼ ω
8281a1i 11 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆m ℕ)) → ℕ ≼ ω)
83 elmapi 8889 . . . . . . . . . 10 (𝑒 ∈ (𝑆m ℕ) → 𝑒:ℕ⟶𝑆)
8483adantl 481 . . . . . . . . 9 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑒:ℕ⟶𝑆)
8584ffvelcdmda 7104 . . . . . . . 8 (((𝜑𝑒 ∈ (𝑆m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒𝑛) ∈ 𝑆)
8680, 82, 85saliuncl 46338 . . . . . . 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 2737 . . . . . 6 ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷)
8978, 79, 87, 88elrestd 45113 . . . . 5 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷))
90 nfra1 3284 . . . . . . . . 9 𝑛𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)
91 rspa 3248 . . . . . . . . 9 ((∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
9290, 91iuneq2df 45052 . . . . . . . 8 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = 𝑛 ∈ ℕ ((𝑒𝑛) ∩ 𝐷))
93 iunin1 5072 . . . . . . . . 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 2777 . . . . . . 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 2835 . . . . 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 3153 . 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cin 3950  c0 4333   ciun 4991   class class class wbr 5143  cmpt 5225  ran crn 5686  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  ωcom 7887  m cmap 8866  cen 8982  cdom 8983  cn 12266  t crest 17465  SAlgcsalg 46323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-ac2 10503  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-acn 9982  df-ac 10156  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-rest 17467  df-salg 46324
This theorem is referenced by:  subsalsal  46374
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