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Theorem subsaliuncl 39052
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 2605 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
2 subsaliuncl.1 . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
31, 2rabexd 4732 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
43ralrimivw 2945 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
5 eqid 2605 . . . . . . . 8 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65fnmpt 5915 . . . . . . 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 10869 . . . . . . 7 ℕ ∈ V
9 fnrndomg 9212 . . . . . . 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 12592 . . . . . 6 ℕ ≈ ω
1312a1i 11 . . . . 5 (𝜑 → ℕ ≈ ω)
14 domentr 7874 . . . . 5 ((ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ℕ ∧ ℕ ≈ ω) → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ω)
1511, 13, 14syl2anc 690 . . . 4 (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ≼ ω)
16 vex 3171 . . . . . . . 8 𝑦 ∈ V
175elrnmpt 5276 . . . . . . . 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 204 . . . . . 6 (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
2019adantl 480 . . . . 5 ((𝜑𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
21 simp3 1055 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
22 subsaliuncl.4 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶𝑇)
2322ffvelrnda 6248 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ 𝑇)
24 subsaliuncl.3 . . . . . . . . . . . . 13 𝑇 = (𝑆t 𝐷)
2523, 24syl6eleq 2693 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (𝑆t 𝐷))
26 subsaliuncl.2 . . . . . . . . . . . . . . 15 (𝜑𝐷𝑉)
2726elexd 3182 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ V)
28 elrest 15853 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
292, 27, 28syl2anc 690 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3029adantr 479 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3125, 30mpbid 220 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
32 rabn0 3907 . . . . . . . . . . 11 ({𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅ ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
3331, 32sylibr 222 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
34333adant3 1073 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
3521, 34eqnetrd 2844 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 ≠ ∅)
36353exp 1255 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅)))
3736rexlimdv 3007 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅))
3837adantr 479 . . . . 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 38210 . . 3 (𝜑 → ∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦))
41 simpl 471 . . . . . 6 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → 𝜑)
42 fveq2 6084 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
4342eqeq1d 2607 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑥𝐷)))
4443rabbidv 3159 . . . . . . . . . . 11 (𝑛 = 𝑚 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4544cbvmptv 4668 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4645rneqi 5256 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4746fneq2i 5882 . . . . . . . 8 (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ↔ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
4847biimpi 204 . . . . . . 7 (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}))
4948ad2antrl 759 . . . . . 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 3114 . . . . . . . . 9 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5150biimpi 204 . . . . . . . 8 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5251adantl 480 . . . . . . 7 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5352adantrl 747 . . . . . 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 1828 . . . . . . 7 𝑧(𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5523ad2ant1 1074 . . . . . . 7 ((𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
56 ineq1 3764 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐷) = (𝑧𝐷))
5756eqeq2d 2615 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑧𝐷)))
5857cbvrabv 3167 . . . . . . . . . 10 {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)} = {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}
5958mpteq2i 4659 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})
6045, 59eqtr2i 2628 . . . . . . . 8 (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6160coeq2i 5188 . . . . . . 7 (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
6247biimpri 216 . . . . . . . 8 (𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) → 𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}))
63623ad2ant2 1075 . . . . . . 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 2614 . . . . . . . . . . 11 ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6564raleqi 3114 . . . . . . . . . 10 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
66 fveq2 6084 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
67 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑦 = 𝑧)
6866, 67eleq12d 2677 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑧) ∈ 𝑧))
6968cbvralv 3142 . . . . . . . . . 10 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
7065, 69bitri 262 . . . . . . . . 9 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
7170biimpi 204 . . . . . . . 8 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
72713ad2ant3 1076 . . . . . . 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 39051 . . . . . 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 1317 . . . . 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 448 . . . 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 1846 . . 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 1074 . . . . . 6 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
79273ad2ant1 1074 . . . . . 6 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝐷 ∈ V)
802adantr 479 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ)) → 𝑆 ∈ SAlg)
81 nnct 12593 . . . . . . . . 9 ℕ ≼ ω
8281a1i 11 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ)) → ℕ ≼ ω)
83 elmapi 7738 . . . . . . . . . 10 (𝑒 ∈ (𝑆𝑚 ℕ) → 𝑒:ℕ⟶𝑆)
8483adantl 480 . . . . . . . . 9 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ)) → 𝑒:ℕ⟶𝑆)
8584ffvelrnda 6248 . . . . . . . 8 (((𝜑𝑒 ∈ (𝑆𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒𝑛) ∈ 𝑆)
8680, 82, 85saliuncl 39018 . . . . . . 7 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ)) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)
87863adant3 1073 . . . . . 6 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)
88 eqid 2605 . . . . . 6 ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷)
8978, 79, 87, 88elrestd 38121 . . . . 5 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷))
90 nfra1 2920 . . . . . . . . 9 𝑛𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)
91 rspa 2909 . . . . . . . . 9 ((∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
9290, 91iuneq2df 38036 . . . . . . . 8 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = 𝑛 ∈ ℕ ((𝑒𝑛) ∩ 𝐷))
93 iunin1 4511 . . . . . . . . 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 2639 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷))
96953ad2ant3 1076 . . . . . 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 2677 . . . . 5 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇 ↔ ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷)))
9989, 98mpbird 245 . . . 4 ((𝜑𝑒 ∈ (𝑆𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)
100993exp 1255 . . 3 (𝜑 → (𝑒 ∈ (𝑆𝑚 ℕ) → (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)))
101100rexlimdv 3007 . 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 194  wa 382  w3a 1030   = wceq 1474  wex 1694  wcel 1975  wne 2775  wral 2891  wrex 2892  {crab 2895  Vcvv 3168  cin 3534  c0 3869   ciun 4445   class class class wbr 4573  cmpt 4633  ran crn 5025  ccom 5028   Fn wfn 5781  wf 5782  cfv 5786  (class class class)co 6523  ωcom 6930  𝑚 cmap 7717  cen 7811  cdom 7812  cn 10863  t crest 15846  SAlgcsalg 39004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-inf2 8394  ax-cc 9113  ax-ac2 9141  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-se 4984  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-isom 5795  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-map 7719  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-card 8621  df-acn 8624  df-ac 8795  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-nn 10864  df-n0 11136  df-z 11207  df-uz 11516  df-rest 15848  df-salg 39005
This theorem is referenced by:  subsalsal  39053
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