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Theorem subsaliuncl 45159
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 2732 . . . . . . . . 9 {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}
2 subsaliuncl.1 . . . . . . . . 9 (𝜑𝑆 ∈ SAlg)
31, 2rabexd 5333 . . . . . . . 8 (𝜑 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
43ralrimivw 3150 . . . . . . 7 (𝜑 → ∀𝑛 ∈ ℕ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ∈ V)
5 eqid 2732 . . . . . . . 8 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
65fnmpt 6690 . . . . . . 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 12220 . . . . . . 7 ℕ ∈ V
9 fnrndomg 10533 . . . . . . 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 13947 . . . . . 6 ℕ ≈ ω
1312a1i 11 . . . . 5 (𝜑 → ℕ ≈ ω)
14 domentr 9011 . . . . 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 3478 . . . . . . . 8 𝑦 ∈ V
175elrnmpt 5955 . . . . . . . 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 482 . . . . 5 ((𝜑𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
21 simp3 1138 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
22 subsaliuncl.4 . . . . . . . . . . . . . 14 (𝜑𝐹:ℕ⟶𝑇)
2322ffvelcdmda 7086 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ 𝑇)
24 subsaliuncl.3 . . . . . . . . . . . . 13 𝑇 = (𝑆t 𝐷)
2523, 24eleqtrdi 2843 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → (𝐹𝑛) ∈ (𝑆t 𝐷))
26 subsaliuncl.2 . . . . . . . . . . . . . . 15 (𝜑𝐷𝑉)
2726elexd 3494 . . . . . . . . . . . . . 14 (𝜑𝐷 ∈ V)
28 elrest 17375 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
292, 27, 28syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3029adantr 481 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝐹𝑛) ∈ (𝑆t 𝐷) ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷)))
3125, 30mpbid 231 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ℕ) → ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
32 rabn0 4385 . . . . . . . . . . 11 ({𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅ ↔ ∃𝑥𝑆 (𝐹𝑛) = (𝑥𝐷))
3331, 32sylibr 233 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
34333adant3 1132 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} ≠ ∅)
3521, 34eqnetrd 3008 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ ∧ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) → 𝑦 ≠ ∅)
36353exp 1119 . . . . . . 7 (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅)))
3736rexlimdv 3153 . . . . . 6 (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} → 𝑦 ≠ ∅))
3837adantr 481 . . . . 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 44006 . . 3 (𝜑 → ∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦))
41 simpl 483 . . . . . 6 ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)) → 𝜑)
42 fveq2 6891 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → (𝐹𝑛) = (𝐹𝑚))
4342eqeq1d 2734 . . . . . . . . . . . 12 (𝑛 = 𝑚 → ((𝐹𝑛) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑥𝐷)))
4443rabbidv 3440 . . . . . . . . . . 11 (𝑛 = 𝑚 → {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)} = {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4544cbvmptv 5261 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4645rneqi 5936 . . . . . . . . 9 ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})
4746fneq2i 6647 . . . . . . . 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 726 . . . . . 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 3323 . . . . . . . . 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 482 . . . . . . 7 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5352adantrl 714 . . . . . 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 1917 . . . . . . 7 𝑧(𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
5523ad2ant1 1133 . . . . . . 7 ((𝜑𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
56 ineq1 4205 . . . . . . . . . . . 12 (𝑥 = 𝑧 → (𝑥𝐷) = (𝑧𝐷))
5756eqeq2d 2743 . . . . . . . . . . 11 (𝑥 = 𝑧 → ((𝐹𝑚) = (𝑥𝐷) ↔ (𝐹𝑚) = (𝑧𝐷)))
5857cbvrabv 3442 . . . . . . . . . 10 {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)} = {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}
5958mpteq2i 5253 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)})
6045, 59eqtr2i 2761 . . . . . . . 8 (𝑚 ∈ ℕ ↦ {𝑧𝑆 ∣ (𝐹𝑚) = (𝑧𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6160coeq2i 5860 . . . . . . 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 1134 . . . . . . 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 2741 . . . . . . . . . . 11 ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})
6564raleqi 3323 . . . . . . . . . 10 (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑚) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦)
66 fveq2 6891 . . . . . . . . . . . 12 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
67 id 22 . . . . . . . . . . . 12 (𝑦 = 𝑧𝑦 = 𝑧)
6866, 67eleq12d 2827 . . . . . . . . . . 11 (𝑦 = 𝑧 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑧) ∈ 𝑧))
6968cbvralvw 3234 . . . . . . . . . 10 (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥𝑆 ∣ (𝐹𝑛) = (𝑥𝐷)})(𝑓𝑧) ∈ 𝑧)
7065, 69bitri 274 . . . . . . . . 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 1135 . . . . . . 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 45158 . . . . . 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 1371 . . . . 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 413 . . . 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 1936 . . 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 1133 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
79273ad2ant1 1133 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝐷 ∈ V)
802adantr 481 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑆 ∈ SAlg)
81 nnct 13948 . . . . . . . . 9 ℕ ≼ ω
8281a1i 11 . . . . . . . 8 ((𝜑𝑒 ∈ (𝑆m ℕ)) → ℕ ≼ ω)
83 elmapi 8845 . . . . . . . . . 10 (𝑒 ∈ (𝑆m ℕ) → 𝑒:ℕ⟶𝑆)
8483adantl 482 . . . . . . . . 9 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑒:ℕ⟶𝑆)
8584ffvelcdmda 7086 . . . . . . . 8 (((𝜑𝑒 ∈ (𝑆m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒𝑛) ∈ 𝑆)
8680, 82, 85saliuncl 45124 . . . . . . 7 ((𝜑𝑒 ∈ (𝑆m ℕ)) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)
87863adant3 1132 . . . . . 6 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝑒𝑛) ∈ 𝑆)
88 eqid 2732 . . . . . 6 ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷)
8978, 79, 87, 88elrestd 43885 . . . . 5 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷))
90 nfra1 3281 . . . . . . . . 9 𝑛𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)
91 rspa 3245 . . . . . . . . 9 ((∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷))
9290, 91iuneq2df 43821 . . . . . . . 8 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = 𝑛 ∈ ℕ ((𝑒𝑛) ∩ 𝐷))
93 iunin1 5075 . . . . . . . . 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 2772 . . . . . . 7 (∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷) → 𝑛 ∈ ℕ (𝐹𝑛) = ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷))
96953ad2ant3 1135 . . . . . 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 2827 . . . . 5 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → ( 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇 ↔ ( 𝑛 ∈ ℕ (𝑒𝑛) ∩ 𝐷) ∈ (𝑆t 𝐷)))
9989, 98mpbird 256 . . . 4 ((𝜑𝑒 ∈ (𝑆m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹𝑛) = ((𝑒𝑛) ∩ 𝐷)) → 𝑛 ∈ ℕ (𝐹𝑛) ∈ 𝑇)
100993exp 1119 . . 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2940  wral 3061  wrex 3070  {crab 3432  Vcvv 3474  cin 3947  c0 4322   ciun 4997   class class class wbr 5148  cmpt 5231  ran crn 5677  ccom 5680   Fn wfn 6538  wf 6539  cfv 6543  (class class class)co 7411  ωcom 7857  m cmap 8822  cen 8938  cdom 8939  cn 12214  t crest 17368  SAlgcsalg 45109
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-ac2 10460  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-int 4951  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-se 5632  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-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-acn 9939  df-ac 10113  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-rest 17370  df-salg 45110
This theorem is referenced by:  subsalsal  45160
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