Theorem subsaliuncllem 46372

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
subsaliuncllem.f	⊢ Ⅎ𝑦𝜑
subsaliuncllem.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
subsaliuncllem.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
subsaliuncllem.e	⊢ 𝐸 = (𝐻 ∘ 𝐺)
subsaliuncllem.h	⊢ (𝜑 → 𝐻 Fn ran 𝐺)
subsaliuncllem.y	⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦)

Assertion

Ref	Expression
subsaliuncllem	⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))

Distinct variable groups:   𝐷,𝑒   𝑥,𝐷   𝑒,𝐸,𝑛   𝑥,𝐸,𝑛   𝑒,𝐹   𝑥,𝐹   𝑦,𝐺   𝑦,𝐻   𝑆,𝑒,𝑛   𝑥,𝑆   𝑦,𝑆,𝑛   𝜑,𝑛

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑒)   𝐷(𝑦,𝑛)   𝐸(𝑦)   𝐹(𝑦,𝑛)   𝐺(𝑥,𝑒,𝑛)   𝐻(𝑥,𝑒,𝑛)   𝑉(𝑥,𝑦,𝑒,𝑛)

Proof of Theorem subsaliuncllem

Step	Hyp	Ref	Expression
1		subsaliuncllem.e	. . 3 ⊢ 𝐸 = (𝐻 ∘ 𝐺)
2		subsaliuncllem.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 Fn ran 𝐺)
3		subsaliuncllem.f	. . . . . . . 8 ⊢ Ⅎ𝑦𝜑
4		vex 3484	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
5		subsaliuncllem.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	elrnmpt 5969	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝐺 ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
7	4, 6	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ran 𝐺 ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
8	7	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran 𝐺 → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
9		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
10		ssrab2 4080	. . . . . . . . . . . . . . . . 17 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ⊆ 𝑆
11	10	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ⊆ 𝑆)
12	9, 11	eqsstrd 4018	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆)
13	12	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆))
14	13	rexlimiv 3148	. . . . . . . . . . . . 13 ⊢ (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆)
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ran 𝐺 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ⊆ 𝑆))
16	8, 15	mpd 15	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ran 𝐺 → 𝑦 ⊆ 𝑆)
17	16	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ⊆ 𝑆)
18		subsaliuncllem.y	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦)
19	18	r19.21bi 3251	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐻‘𝑦) ∈ 𝑦)
20	17, 19	sseldd 3984	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐻‘𝑦) ∈ 𝑆)
21	20	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻‘𝑦) ∈ 𝑆))
22	3, 21	ralrimi 3257	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆)
23	2, 22	jca 511	. . . . . 6 ⊢ (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆))
24		ffnfv 7139	. . . . . 6 ⊢ (𝐻:ran 𝐺⟶𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆))
25	23, 24	sylibr 234	. . . . 5 ⊢ (𝜑 → 𝐻:ran 𝐺⟶𝑆)
26		eqid 2737	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}
27		subsaliuncllem.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
28	26, 27	rabexd 5340	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
29	28	ralrimivw 3150	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
30	5	fnmpt 6708	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V → 𝐺 Fn ℕ)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn ℕ)
32		dffn3 6748	. . . . . 6 ⊢ (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
33	31, 32	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶ran 𝐺)
34		fco 6760	. . . . 5 ⊢ ((𝐻:ran 𝐺⟶𝑆 ∧ 𝐺:ℕ⟶ran 𝐺) → (𝐻 ∘ 𝐺):ℕ⟶𝑆)
35	25, 33, 34	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐺):ℕ⟶𝑆)
36		nnex 12272	. . . . . 6 ⊢ ℕ ∈ V
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
38	27, 37	elmapd 8880	. . . 4 ⊢ (𝜑 → ((𝐻 ∘ 𝐺) ∈ (𝑆 ↑m ℕ) ↔ (𝐻 ∘ 𝐺):ℕ⟶𝑆))
39	35, 38	mpbird 257	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝐺) ∈ (𝑆 ↑m ℕ))
40	1, 39	eqeltrid 2845	. 2 ⊢ (𝜑 → 𝐸 ∈ (𝑆 ↑m ℕ))
41	33	ffvelcdmda 7104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ran 𝐺)
42	18	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦)
43		fveq2 6906	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑛) → (𝐻‘𝑦) = (𝐻‘(𝐺‘𝑛)))
44		id 22	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑛) → 𝑦 = (𝐺‘𝑛))
45	43, 44	eleq12d 2835	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑛) → ((𝐻‘𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛)))
46	45	rspcva 3620	. . . . . . 7 ⊢ (((𝐺‘𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) → (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛))
47	41, 42, 46	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛))
48	33	ffund 6740	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
49	48	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun 𝐺)
50		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
51	5	dmeqi 5915	. . . . . . . . . . . . 13 ⊢ dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
52	51	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → dom 𝐺 = dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
53		dmmptg 6262	. . . . . . . . . . . . 13 ⊢ (∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V → dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ℕ)
54	29, 53	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → dom (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ℕ)
55	52, 54	eqtrd 2777	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = ℕ)
56	55	eqcomd 2743	. . . . . . . . . 10 ⊢ (𝜑 → ℕ = dom 𝐺)
57	56	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom 𝐺)
58	50, 57	eleqtrd 2843	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
59	49, 58, 1	fvcod 45232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) = (𝐻‘(𝐺‘𝑛)))
60	5	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
61	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
62	60, 61	fvmpt2d 7029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
63	62	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = (𝐺‘𝑛))
64	59, 63	eleq12d 2835	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ↔ (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛)))
65	47, 64	mpbird 257	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
66		ineq1 4213	. . . . . . 7 ⊢ (𝑥 = (𝐸‘𝑛) → (𝑥 ∩ 𝐷) = ((𝐸‘𝑛) ∩ 𝐷))
67	66	eqeq2d 2748	. . . . . 6 ⊢ (𝑥 = (𝐸‘𝑛) → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
68	67	elrab 3692	. . . . 5 ⊢ ((𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ↔ ((𝐸‘𝑛) ∈ 𝑆 ∧ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
69	65, 68	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ 𝑆 ∧ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
70	69	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))
71	70	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))
72		fveq1 6905	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝑒‘𝑛) = (𝐸‘𝑛))
73	72	ineq1d 4219	. . . . 5 ⊢ (𝑒 = 𝐸 → ((𝑒‘𝑛) ∩ 𝐷) = ((𝐸‘𝑛) ∩ 𝐷))
74	73	eqeq2d 2748	. . . 4 ⊢ (𝑒 = 𝐸 → ((𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ↔ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
75	74	ralbidv 3178	. . 3 ⊢ (𝑒 = 𝐸 → (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ↔ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
76	75	rspcev 3622	. 2 ⊢ ((𝐸 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)) → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
77	40, 71, 76	syl2anc 584	1 ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951   ↦ cmpt 5225  dom cdm 5685  ran crn 5686   ∘ ccom 5689  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  ℕcn 12266

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-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215

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-ral 3062  df-rex 3071  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-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-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-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-map 8868  df-nn 12267

This theorem is referenced by:  subsaliuncl  46373