Theorem subsaliuncl 44589

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	⊢ (𝜑 → 𝑆 ∈ SAlg)
subsaliuncl.2	⊢ (𝜑 → 𝐷 ∈ 𝑉)
subsaliuncl.3	⊢ 𝑇 = (𝑆 ↾t 𝐷)
subsaliuncl.4	⊢ (𝜑 → 𝐹:ℕ⟶𝑇)

Assertion

Ref	Expression
subsaliuncl	⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 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.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}
2		subsaliuncl.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3	1, 2	rabexd 5290	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
4	3	ralrimivw 3147	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
5		eqid 2736	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	fnmpt 6641	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V → (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ)
7	4, 6	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ)
8		nnex 12159	. . . . . . 7 ⊢ ℕ ∈ V
9		fnrndomg 10472	. . . . . . 7 ⊢ (ℕ ∈ V → ((𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ))
10	8, 9	ax-mp 5	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ)
11	7, 10	syl 17	. . . . 5 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ)
12		nnenom 13885	. . . . . 6 ⊢ ℕ ≈ ω
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ≈ ω)
14		domentr 8953	. . . . 5 ⊢ ((ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ ∧ ℕ ≈ ω) → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ω)
15	11, 13, 14	syl2anc 584	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ω)
16		vex 3449	. . . . . . . 8 ⊢ 𝑦 ∈ V
17	5	elrnmpt 5911	. . . . . . . 8 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
18	16, 17	ax-mp 5	. . . . . . 7 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
19	18	biimpi 215	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
20	19	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
21		simp3 1138	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
22		subsaliuncl.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℕ⟶𝑇)
23	22	ffvelcdmda 7035	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑇)
24		subsaliuncl.3	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑆 ↾t 𝐷)
25	23, 24	eleqtrdi 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷))
26		subsaliuncl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27	26	elexd 3465	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ V)
28		elrest 17309	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
29	2, 27, 28	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
30	29	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
31	25, 30	mpbid 231	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
32		rabn0 4345	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
33	31, 32	sylibr 233	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
34	33	3adant3 1132	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
35	21, 34	eqnetrd 3011	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 ≠ ∅)
36	35	3exp 1119	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅)))
37	36	rexlimdv 3150	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
39	20, 38	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → 𝑦 ≠ ∅)
40	15, 39	axccdom 43433	. . 3 ⊢ (𝜑 → ∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦))
41		simpl 483	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)) → 𝜑)
42		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
43	42	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)))
44	43	rabbidv 3415	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
45	44	cbvmptv 5218	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
46	45	rneqi 5892	. . . . . . . . 9 ⊢ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
47	46	fneq2i 6600	. . . . . . . 8 ⊢ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
48	47	biimpi 215	. . . . . . 7 ⊢ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
49	48	ad2antrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
50	46	raleqi 3311	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)
51	50	biimpi 215	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦 → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)
52	51	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)
53	52	adantrl 714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)) → ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)
54		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑧(𝜑 ∧ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)
55	2	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
56		ineq1 4165	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝐷) = (𝑧 ∩ 𝐷))
57	56	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)))
58	57	cbvrabv 3417	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)} = {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}
59	58	mpteq2i 5210	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})
60	45, 59	eqtr2i 2765	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
61	60	coeq2i 5816	. . . . . . 7 ⊢ (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
62	47	biimpri 227	. . . . . . . 8 ⊢ (𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) → 𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
63	62	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → 𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
64	46	eqcomi 2745	. . . . . . . . . . 11 ⊢ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
65	64	raleqi 3311	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)
66		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
67		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
68	66, 67	eleq12d 2832	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑧) ∈ 𝑧))
69	68	cbvralvw 3225	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑧) ∈ 𝑧)
70	65, 69	bitri 274	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦 ↔ ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑧) ∈ 𝑧)
71	70	biimpi 215	. . . . . . . 8 ⊢ (∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦 → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑧) ∈ 𝑧)
72	71	3ad2ant3 1135	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → ∀𝑧 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑧) ∈ 𝑧)
73	54, 55, 5, 61, 63, 72	subsaliuncllem 44588	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
74	41, 49, 53, 73	syl3anc 1371	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦)) → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
75	74	ex 413	. . . 4 ⊢ (𝜑 → ((𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)))
76	75	exlimdv 1936	. . 3 ⊢ (𝜑 → (∃𝑓(𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ∧ ∀𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})(𝑓‘𝑦) ∈ 𝑦) → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)))
77	40, 76	mpd 15	. 2 ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
78	2	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
79	27	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝐷 ∈ V)
80	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑆 ∈ SAlg)
81		nnct 13886	. . . . . . . . 9 ⊢ ℕ ≼ ω
82	81	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ℕ ≼ ω)
83		elmapi 8787	. . . . . . . . . 10 ⊢ (𝑒 ∈ (𝑆 ↑m ℕ) → 𝑒:ℕ⟶𝑆)
84	83	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑒:ℕ⟶𝑆)
85	84	ffvelcdmda 7035	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ 𝑆)
86	80, 82, 85	saliuncl 44554	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
87	86	3adant3 1132	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
88		eqid 2736	. . . . . 6 ⊢ (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
89	78, 79, 87, 88	elrestd 43308	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
90		nfra1 3267	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)
91		rspa 3231	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
92	90, 91	iuneq2df 43244	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷))
93		iunin1 5032	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
94	93	a1i 11	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
95	92, 94	eqtrd 2776	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
96	95	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
97	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑇 = (𝑆 ↾t 𝐷))
98	96, 97	eleq12d 2832	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → (∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇 ↔ (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷)))
99	89, 98	mpbird 256	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)
100	99	3exp 1119	. . 3 ⊢ (𝜑 → (𝑒 ∈ (𝑆 ↑m ℕ) → (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)))
101	100	rexlimdv 3150	. 2 ⊢ (𝜑 → (∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇))
102	77, 101	mpd 15	1 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ∩ cin 3909  ∅c0 4282  ∪ ciun 4954   class class class wbr 5105   ↦ cmpt 5188  ran crn 5634   ∘ ccom 5637   Fn wfn 6491  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  ωcom 7802   ↑_m cmap 8765   ≈ cen 8880   ≼ cdom 8881  ℕcn 12153   ↾_t crest 17302  SAlgcsalg 44539

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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-rest 17304  df-salg 44540

This theorem is referenced by:  subsalsal  44590