Theorem subsaliuncl 40576

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 𝑥 𝑒 𝑓 𝑧 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}
2		subsaliuncl.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3	1, 2	rabexd 4814	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
4	3	ralrimivw 2967	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
5		eqid 2622	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	fnmpt 6020	. . . . . . 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 11026	. . . . . . 7 ⊢ ℕ ∈ V
9		fnrndomg 9358	. . . . . . 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 12779	. . . . . 6 ⊢ ℕ ≈ ω
13	12	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ≈ ω)
14		domentr 8015	. . . . 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 693	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ω)
16		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
17	5	elrnmpt 5372	. . . . . . . 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 206	. . . . . 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 1063	. . . . . . . . 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	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑇)
24		subsaliuncl.3	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑆 ↾t 𝐷)
25	23, 24	syl6eleq 2711	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷))
26		subsaliuncl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
27	26	elexd 3214	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ V)
28		elrest 16088	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
29	2, 27, 28	syl2anc 693	. . . . . . . . . . . . 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 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
32		rabn0 3958	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
33	31, 32	sylibr 224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
34	33	3adant3 1081	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
35	21, 34	eqnetrd 2861	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 ≠ ∅)
36	35	3exp 1264	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅)))
37	36	rexlimdv 3030	. . . . . 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 39416	. . 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 473	. . . . . 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 6191	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
43	42	eqeq1d 2624	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)))
44	43	rabbidv 3189	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
45	44	cbvmptv 4750	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
46	45	rneqi 5352	. . . . . . . . 9 ⊢ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
47	46	fneq2i 5986	. . . . . . . 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 206	. . . . . . 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 764	. . . . . 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 3142	. . . . . . . . 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 206	. . . . . . . 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 752	. . . . . 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 1843	. . . . . . 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 1082	. . . . . . 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 3807	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝐷) = (𝑧 ∩ 𝐷))
57	56	eqeq2d 2632	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)))
58	57	cbvrabv 3199	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)} = {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}
59	58	mpteq2i 4741	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})
60	45, 59	eqtr2i 2645	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
61	60	coeq2i 5282	. . . . . . 7 ⊢ (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
62	47	biimpri 218	. . . . . . . 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 1083	. . . . . . 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 2631	. . . . . . . . . . 11 ⊢ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
65	64	raleqi 3142	. . . . . . . . . 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 6191	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
67		id 22	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
68	66, 67	eleq12d 2695	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑧) ∈ 𝑧))
69	68	cbvralv 3171	. . . . . . . . . 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 264	. . . . . . . . 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 206	. . . . . . . 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 1084	. . . . . . 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 40575	. . . . . 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 1326	. . . . 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 450	. . . 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 1861	. . 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 1082	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
79	27	3ad2ant1 1082	. . . . . 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 12780	. . . . . . . . 9 ⊢ ℕ ≼ ω
82	81	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ)) → ℕ ≼ ω)
83		elmapi 7879	. . . . . . . . . 10 ⊢ (𝑒 ∈ (𝑆 ↑𝑚 ℕ) → 𝑒:ℕ⟶𝑆)
84	83	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ)) → 𝑒:ℕ⟶𝑆)
85	84	ffvelrnda 6359	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ 𝑆)
86	80, 82, 85	saliuncl 40542	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
87	86	3adant3 1081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
88		eqid 2622	. . . . . 6 ⊢ (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
89	78, 79, 87, 88	elrestd 39291	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
90		nfra1 2941	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)
91		rspa 2930	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
92	90, 91	iuneq2df 39212	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷))
93		iunin1 4585	. . . . . . . . 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 2656	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
96	95	3ad2ant3 1084	. . . . . 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 2695	. . . . 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 247	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑𝑚 ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)
100	99	3exp 1264	. . 3 ⊢ (𝜑 → (𝑒 ∈ (𝑆 ↑𝑚 ℕ) → (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)))
101	100	rexlimdv 3030	. 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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∩ cin 3573  ∅c0 3915  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  ran crn 5115   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065   ↑_𝑚 cmap 7857   ≈ cen 7952   ≼ cdom 7953  ℕcn 11020   ↾_t crest 16081  SAlgcsalg 40528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rest 16083  df-salg 40529

This theorem is referenced by:  subsalsal  40577