Theorem subsaliuncl 46387

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 2735	. . . . . . . . 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 3136	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
5		eqid 2735	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	fnmpt 6678	. . . . . . 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 12246	. . . . . . 7 ⊢ ℕ ∈ V
9		fnrndomg 10550	. . . . . . 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 13998	. . . . . 6 ⊢ ℕ ≈ ω
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ≈ ω)
14		domentr 9027	. . . . 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 3463	. . . . . . . 8 ⊢ 𝑦 ∈ V
17	5	elrnmpt 5938	. . . . . . . 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 216	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
20	19	adantl 481	. . . . 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 7074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑇)
24		subsaliuncl.3	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑆 ↾t 𝐷)
25	23, 24	eleqtrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷))
26		subsaliuncl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27	26	elexd 3483	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ V)
28		elrest 17441	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
31	25, 30	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
32		rabn0 4364	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
33	31, 32	sylibr 234	. . . . . . . . . 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 2999	. . . . . . . 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 3139	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
38	37	adantr 480	. . . . 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 45246	. . 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 482	. . . . . 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 6876	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
43	42	eqeq1d 2737	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)))
44	43	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
45	44	cbvmptv 5225	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
46	45	rneqi 5917	. . . . . . . . 9 ⊢ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
47	46	fneq2i 6636	. . . . . . . 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 216	. . . . . . 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 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)) → 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
50	46	raleqi 3303	. . . . . . . . 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 216	. . . . . . . 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 481	. . . . . . 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 716	. . . . . 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 1914	. . . . . . 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 4188	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝐷) = (𝑧 ∩ 𝐷))
57	56	eqeq2d 2746	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)))
58	57	cbvrabv 3426	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)} = {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}
59	58	mpteq2i 5217	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})
60	45, 59	eqtr2i 2759	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
61	60	coeq2i 5840	. . . . . . 7 ⊢ (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
62	47	biimpri 228	. . . . . . . 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 2744	. . . . . . . . . . 11 ⊢ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
65	64	raleqi 3303	. . . . . . . . . 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 6876	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
67		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
68	66, 67	eleq12d 2828	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑧) ∈ 𝑧))
69	68	cbvralvw 3220	. . . . . . . . . 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 275	. . . . . . . . 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 216	. . . . . . . 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 46386	. . . . . 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 1373	. . . . 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 412	. . . 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 1933	. . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑆 ∈ SAlg)
81		nnct 13999	. . . . . . . . 9 ⊢ ℕ ≼ ω
82	81	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ℕ ≼ ω)
83		elmapi 8863	. . . . . . . . . 10 ⊢ (𝑒 ∈ (𝑆 ↑m ℕ) → 𝑒:ℕ⟶𝑆)
84	83	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑒:ℕ⟶𝑆)
85	84	ffvelcdmda 7074	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ 𝑆)
86	80, 82, 85	saliuncl 46352	. . . . . . 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 2735	. . . . . 6 ⊢ (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
89	78, 79, 87, 88	elrestd 45132	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
90		nfra1 3266	. . . . . . . . 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 45071	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷))
93		iunin1 5048	. . . . . . . . 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 2770	. . . . . . 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 2828	. . . . 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 257	. . . 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 3139	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459   ∩ cin 3925  ∅c0 4308  ∪ ciun 4967   class class class wbr 5119   ↦ cmpt 5201  ran crn 5655   ∘ ccom 5658   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  ωcom 7861   ↑_m cmap 8840   ≈ cen 8956   ≼ cdom 8957  ℕcn 12240   ↾_t crest 17434  SAlgcsalg 46337

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cc 10449  ax-ac2 10477  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-card 9953  df-acn 9956  df-ac 10130  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-rest 17436  df-salg 46338

This theorem is referenced by:  subsalsal  46388