Theorem subsaliuncl 47092

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 2763	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}
2		subsaliuncl.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3	1, 2	rabexd 5310	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
4	3	ralrimivw 3161	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
5		eqid 2763	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	fnmpt 6675	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V → (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ)
7	4, 6	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ)
8		nnex 12234	. . . . . . 7 ⊢ ℕ ∈ V
9		fnrndomg 10515	. . . . . . 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 18	. . . . 5 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ)
12		nnenom 14012	. . . . . 6 ⊢ ℕ ≈ ω
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ≈ ω)
14		domentr 9006	. . . . 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 595	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ω)
16		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
17	5	elrnmpt 5948	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
19	18	bilani 509	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
20		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
21		subsaliuncl.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℕ⟶𝑇)
22	21	ffvelcdmda 7079	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑇)
23		subsaliuncl.3	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑆 ↾t 𝐷)
24	22, 23	eleqtrdi 2873	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷))
25		subsaliuncl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
26	25	elexd 3478	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ V)
27		elrest 17475	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
28	2, 26, 27	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
29	28	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
30	24, 29	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
31		rabn0 4346	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
32	30, 31	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
33	32	3adant3 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
34	20, 33	eqnetrd 3025	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 ≠ ∅)
35	34	3exp 1137	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅)))
36	35	rexlimdv 3164	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
37	36	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
38	19, 37	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → 𝑦 ≠ ∅)
39	15, 38	axccdom 45958	. . 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))
40		simpl 487	. . . . . 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)
41		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
42	41	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)))
43	42	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
44	43	cbvmptv 5215	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
45	44	rneqi 5927	. . . . . . . . 9 ⊢ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
46	45	fneq2i 6633	. . . . . . . 8 ⊢ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
47	46	biimpi 219	. . . . . . 7 ⊢ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
48	47	ad2antrl 740	. . . . . 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)}))
49	45	raleqi 3321	. . . . . . . 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)
50	49	bilani 509	. . . . . . 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)
51	50	adantrl 728	. . . . . 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)
52		nfv 1944	. . . . . . 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)
53	2	3ad2ant1 1151	. . . . . . 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)
54		ineq1 4166	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝐷) = (𝑧 ∩ 𝐷))
55	54	eqeq2d 2774	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)))
56	55	cbvrabv 3426	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)} = {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}
57	56	mpteq2i 5207	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})
58	44, 57	eqtr2i 2787	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
59	58	coeq2i 5846	. . . . . . 7 ⊢ (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
60	46	biimpri 231	. . . . . . . 8 ⊢ (𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) → 𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
61	60	3ad2ant2 1152	. . . . . . 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)}))
62	45	eqcomi 2772	. . . . . . . . . . 11 ⊢ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
63	62	raleqi 3321	. . . . . . . . . 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)
64		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
65		id 23	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
66	64, 65	eleq12d 2857	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑧) ∈ 𝑧))
67	66	cbvralvw 3243	. . . . . . . . . 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)
68	63, 67	bitri 278	. . . . . . . . 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)
69	68	biimpi 219	. . . . . . . 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)
70	69	3ad2ant3 1153	. . . . . . 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)
71	52, 53, 5, 59, 61, 70	subsaliuncllem 47091	. . . . . 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))
72	40, 48, 51, 71	syl3anc 1398	. . . . 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))
73	72	ex 417	. . . 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)))
74	73	exlimdv 1963	. . 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)))
75	39, 74	mpd 16	. 2 ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
76	2	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
77	26	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝐷 ∈ V)
78	2	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑆 ∈ SAlg)
79		nnct 14013	. . . . . . . . 9 ⊢ ℕ ≼ ω
80	79	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ℕ ≼ ω)
81		elmapi 8842	. . . . . . . . . 10 ⊢ (𝑒 ∈ (𝑆 ↑m ℕ) → 𝑒:ℕ⟶𝑆)
82	81	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑒:ℕ⟶𝑆)
83	82	ffvelcdmda 7079	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ 𝑆)
84	78, 80, 83	saliuncl 47057	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
85	84	3adant3 1150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
86		eqid 2763	. . . . . 6 ⊢ (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
87	76, 77, 85, 86	elrestd 45846	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
88		nfra1 3289	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)
89		rspa 3254	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
90	88, 89	iuneq2df 45787	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷))
91		iunin1 5036	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
92	91	a1i 11	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
93	90, 92	eqtrd 2798	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
94	93	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
95	23	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑇 = (𝑆 ↾t 𝐷))
96	94, 95	eleq12d 2857	. . . . 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)))
97	87, 96	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)
98	97	3exp 1137	. . 3 ⊢ (𝜑 → (𝑒 ∈ (𝑆 ↑m ℕ) → (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)))
99	98	rexlimdv 3164	. 2 ⊢ (𝜑 → (∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇))
100	75, 99	mpd 16	1 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∩ cin 3904  ∅c0 4286  ∪ ciun 4956   class class class wbr 5109   ↦ cmpt 5192  ran crn 5662   ∘ ccom 5665   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ωcom 7858   ↑_m cmap 8820   ≈ cen 8936   ≼ cdom 8937  ℕcn 12228   ↾_t crest 17468  SAlgcsalg 47042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-inf2 9606  ax-cc 10414  ax-ac2 10442  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-acn 9924  df-ac 10096  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-rest 17470  df-salg 47043

This theorem is referenced by:  subsalsal  47093