Theorem subsaliuncllem 45073

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 3479	. . . . . . . . . . . . . 14 ⊢ 𝑦 ∈ V
5		subsaliuncllem.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	elrnmpt 5956	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . 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 4078	. . . . . . . . . . . . . . . . 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 4021	. . . . . . . . . . . . . . 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 3149	. . . . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → 𝑦 ⊆ 𝑆)
18		subsaliuncllem.y	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦)
19	18	r19.21bi 3249	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐻‘𝑦) ∈ 𝑦)
20	17, 19	sseldd 3984	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝐺) → (𝐻‘𝑦) ∈ 𝑆)
21	20	ex 414	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ ran 𝐺 → (𝐻‘𝑦) ∈ 𝑆))
22	3, 21	ralrimi 3255	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆)
23	2, 22	jca 513	. . . . . 6 ⊢ (𝜑 → (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆))
24		ffnfv 7118	. . . . . 6 ⊢ (𝐻:ran 𝐺⟶𝑆 ↔ (𝐻 Fn ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑆))
25	23, 24	sylibr 233	. . . . 5 ⊢ (𝜑 → 𝐻:ran 𝐺⟶𝑆)
26		eqid 2733	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}
27		subsaliuncllem.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
28	26, 27	rabexd 5334	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
29	28	ralrimivw 3151	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
30	5	fnmpt 6691	. . . . . . 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 6731	. . . . . 6 ⊢ (𝐺 Fn ℕ ↔ 𝐺:ℕ⟶ran 𝐺)
33	31, 32	sylib 217	. . . . 5 ⊢ (𝜑 → 𝐺:ℕ⟶ran 𝐺)
34		fco 6742	. . . . 5 ⊢ ((𝐻:ran 𝐺⟶𝑆 ∧ 𝐺:ℕ⟶ran 𝐺) → (𝐻 ∘ 𝐺):ℕ⟶𝑆)
35	25, 33, 34	syl2anc 585	. . . 4 ⊢ (𝜑 → (𝐻 ∘ 𝐺):ℕ⟶𝑆)
36		nnex 12218	. . . . . 6 ⊢ ℕ ∈ V
37	36	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ∈ V)
38	27, 37	elmapd 8834	. . . 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 2838	. 2 ⊢ (𝜑 → 𝐸 ∈ (𝑆 ↑m ℕ))
41	33	ffvelcdmda 7087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ran 𝐺)
42	18	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦)
43		fveq2 6892	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑛) → (𝐻‘𝑦) = (𝐻‘(𝐺‘𝑛)))
44		id 22	. . . . . . . . 9 ⊢ (𝑦 = (𝐺‘𝑛) → 𝑦 = (𝐺‘𝑛))
45	43, 44	eleq12d 2828	. . . . . . . 8 ⊢ (𝑦 = (𝐺‘𝑛) → ((𝐻‘𝑦) ∈ 𝑦 ↔ (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛)))
46	45	rspcva 3611	. . . . . . 7 ⊢ (((𝐺‘𝑛) ∈ ran 𝐺 ∧ ∀𝑦 ∈ ran 𝐺(𝐻‘𝑦) ∈ 𝑦) → (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛))
47	41, 42, 46	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐻‘(𝐺‘𝑛)) ∈ (𝐺‘𝑛))
48	33	ffund 6722	. . . . . . . . 9 ⊢ (𝜑 → Fun 𝐺)
49	48	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → Fun 𝐺)
50		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
51	5	dmeqi 5905	. . . . . . . . . . . . 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 6242	. . . . . . . . . . . . 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 2773	. . . . . . . . . . 11 ⊢ (𝜑 → dom 𝐺 = ℕ)
56	55	eqcomd 2739	. . . . . . . . . 10 ⊢ (𝜑 → ℕ = dom 𝐺)
57	56	adantr 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ℕ = dom 𝐺)
58	50, 57	eleqtrd 2836	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ dom 𝐺)
59	49, 58, 1	fvcod 43926	. . . . . . 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 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
62	60, 61	fvmpt2d 7012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
63	62	eqcomd 2739	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = (𝐺‘𝑛))
64	59, 63	eleq12d 2828	. . . . . 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 4206	. . . . . . 7 ⊢ (𝑥 = (𝐸‘𝑛) → (𝑥 ∩ 𝐷) = ((𝐸‘𝑛) ∩ 𝐷))
67	66	eqeq2d 2744	. . . . . 6 ⊢ (𝑥 = (𝐸‘𝑛) → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
68	67	elrab 3684	. . . . 5 ⊢ ((𝐸‘𝑛) ∈ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ↔ ((𝐸‘𝑛) ∈ 𝑆 ∧ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
69	65, 68	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐸‘𝑛) ∈ 𝑆 ∧ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷)))
70	69	simprd 497	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))
71	70	ralrimiva 3147	. 2 ⊢ (𝜑 → ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝐸‘𝑛) ∩ 𝐷))
72		fveq1 6891	. . . . . 6 ⊢ (𝑒 = 𝐸 → (𝑒‘𝑛) = (𝐸‘𝑛))
73	72	ineq1d 4212	. . . . 5 ⊢ (𝑒 = 𝐸 → ((𝑒‘𝑛) ∩ 𝐷) = ((𝐸‘𝑛) ∩ 𝐷))
74	73	eqeq2d 2744	. . . 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 3613	. 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 585	1 ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  Ⅎwnf 1786   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   ↦ cmpt 5232  dom cdm 5677  ran crn 5678   ∘ ccom 5681  Fun wfun 6538   Fn wfn 6539  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409   ↑_m cmap 8820  ℕcn 12212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-map 8822  df-nn 12213

This theorem is referenced by:  subsaliuncl  45074