Theorem subsaliuncl 47367

Description: A subspace sigma-algebra is closed under countable union. This is Lemma 121A (iii) of [Fremlin1] p. 35. The proof uses fnrndomnum 10617 rather than fnrndomg 10618, and so does not require ax-ac 10537. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Vincent Gonzalez, 30-Aug-2026.)

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 2761	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}
2		subsaliuncl.1	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3	1, 2	rabexd 5301	. . . . . . . 8 ⊢ (𝜑 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
4	3	ralrimivw 3159	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ℕ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ∈ V)
5		eqid 2761	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
6	5	fnmpt 6679	. . . . . . 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		omelon 9647	. . . . . . . 8 ⊢ ω ∈ On
9		nnct 14124	. . . . . . . 8 ⊢ ℕ ≼ ω
10		ondomen 10116	. . . . . . . 8 ⊢ ((ω ∈ On ∧ ℕ ≼ ω) → ℕ ∈ dom card)
11	8, 9, 10	mp2an 705	. . . . . . 7 ⊢ ℕ ∈ dom card
12		fnrndomnum 10617	. . . . . . 7 ⊢ (ℕ ∈ dom card → ((𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ))
13	11, 12	ax-mp 5	. . . . . 6 ⊢ ((𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) Fn ℕ → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ)
14	7, 13	syl 18	. . . . 5 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ)
15	9	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ≼ ω)
16		domtr 9034	. . . . 5 ⊢ ((ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ℕ ∧ ℕ ≼ ω) → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ω)
17	14, 15, 16	syl2anc 596	. . . 4 ⊢ (𝜑 → ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ≼ ω)
18		vex 3455	. . . . . . 7 ⊢ 𝑦 ∈ V
19	5	elrnmpt 5940	. . . . . . 7 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
20	18, 19	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
21	20	bilani 510	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → ∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
22		simp3 1156	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
23		subsaliuncl.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:ℕ⟶𝑇)
24	23	ffvelcdmda 7084	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ 𝑇)
25		subsaliuncl.3	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑆 ↾t 𝐷)
26	24, 25	eleqtrdi 2871	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷))
27		subsaliuncl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
28	27	elexd 3474	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ V)
29		elrest 17598	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ SAlg ∧ 𝐷 ∈ V) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
30	2, 28, 29	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
31	30	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) ∈ (𝑆 ↾t 𝐷) ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷)))
32	26, 31	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
33		rabn0 4339	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅ ↔ ∃𝑥 ∈ 𝑆 (𝐹‘𝑛) = (𝑥 ∩ 𝐷))
34	32, 33	sylibr 237	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
35	34	3adant3 1150	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} ≠ ∅)
36	22, 35	eqnetrd 3023	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ ∧ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) → 𝑦 ≠ ∅)
37	36	3exp 1137	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ℕ → (𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅)))
38	37	rexlimdv 3162	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
39	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → (∃𝑛 ∈ ℕ 𝑦 = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} → 𝑦 ≠ ∅))
40	21, 39	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})) → 𝑦 ≠ ∅)
41	17, 40	axccdom 46234	. . 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))
42		simpl 488	. . . . . 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)
43		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
44	43	eqeq1d 2763	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → ((𝐹‘𝑛) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)))
45	44	rabbidv 3420	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)} = {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
46	45	cbvmptv 5209	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
47	46	rneqi 5919	. . . . . . . . 9 ⊢ ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) = ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)})
48	47	fneq2i 6637	. . . . . . . 8 ⊢ (𝑓 Fn ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}) ↔ 𝑓 Fn ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}))
49	48	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)}))
50	49	ad2antrl 741	. . . . . 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)}))
51	47	raleqi 3318	. . . . . . . 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	bilani 510	. . . . . . 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 729	. . . . . 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 1947	. . . . . . 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 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)
56		ineq1 4159	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥 ∩ 𝐷) = (𝑧 ∩ 𝐷))
57	56	eqeq2d 2772	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑚) = (𝑥 ∩ 𝐷) ↔ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)))
58	57	cbvrabv 3423	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)} = {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}
59	58	mpteq2i 5201	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})
60	46, 59	eqtr2i 2785	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)}) = (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
61	60	coeq2i 5838	. . . . . . 7 ⊢ (𝑓 ∘ (𝑚 ∈ ℕ ↦ {𝑧 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑧 ∩ 𝐷)})) = (𝑓 ∘ (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)}))
62	48	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)}))
63	62	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)}))
64	47	eqcomi 2770	. . . . . . . . . . 11 ⊢ ran (𝑚 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑚) = (𝑥 ∩ 𝐷)}) = ran (𝑛 ∈ ℕ ↦ {𝑥 ∈ 𝑆 ∣ (𝐹‘𝑛) = (𝑥 ∩ 𝐷)})
65	64	raleqi 3318	. . . . . . . . . 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 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
67		id 23	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → 𝑦 = 𝑧)
68	66, 67	eleq12d 2855	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ∈ 𝑦 ↔ (𝑓‘𝑧) ∈ 𝑧))
69	68	cbvralvw 3241	. . . . . . . . . 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 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)
71	70	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)
72	71	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)
73	54, 55, 5, 61, 63, 72	subsaliuncllem 47366	. . . . . 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	42, 50, 53, 73	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))
75	74	ex 418	. . . 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 1966	. . 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	41, 76	mpd 16	. 2 ⊢ (𝜑 → ∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
78	2	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑆 ∈ SAlg)
79	28	3ad2ant1 1151	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝐷 ∈ V)
80	2	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑆 ∈ SAlg)
81	9	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ℕ ≼ ω)
82		elmapi 8869	. . . . . . . . . 10 ⊢ (𝑒 ∈ (𝑆 ↑m ℕ) → 𝑒:ℕ⟶𝑆)
83	82	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → 𝑒:ℕ⟶𝑆)
84	83	ffvelcdmda 7084	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) ∧ 𝑛 ∈ ℕ) → (𝑒‘𝑛) ∈ 𝑆)
85	80, 81, 84	saliuncl 47332	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
86	85	3adant3 1150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∈ 𝑆)
87		eqid 2761	. . . . . 6 ⊢ (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
88	78, 79, 86, 87	elrestd 46122	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷) ∈ (𝑆 ↾t 𝐷))
89		nfra1 3287	. . . . . . . . 9 ⊢ Ⅎ𝑛∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)
90		rspa 3252	. . . . . . . . 9 ⊢ ((∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷))
91	89, 90	iuneq2df 46063	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷))
92		iunin1 5030	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷)
93	92	a1i 11	. . . . . . . 8 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ ((𝑒‘𝑛) ∩ 𝐷) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
94	91, 93	eqtrd 2796	. . . . . . 7 ⊢ (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
95	94	3ad2ant3 1153	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) = (∪ 𝑛 ∈ ℕ (𝑒‘𝑛) ∩ 𝐷))
96	25	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → 𝑇 = (𝑆 ↾t 𝐷))
97	95, 96	eleq12d 2855	. . . . 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)))
98	88, 97	mpbird 260	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝑆 ↑m ℕ) ∧ ∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷)) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)
99	98	3exp 1137	. . 3 ⊢ (𝜑 → (𝑒 ∈ (𝑆 ↑m ℕ) → (∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)))
100	99	rexlimdv 3162	. 2 ⊢ (𝜑 → (∃𝑒 ∈ (𝑆 ↑m ℕ)∀𝑛 ∈ ℕ (𝐹‘𝑛) = ((𝑒‘𝑛) ∩ 𝐷) → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇))
101	77, 100	mpd 16	1 ⊢ (𝜑 → ∪ 𝑛 ∈ ℕ (𝐹‘𝑛) ∈ 𝑇)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ∩ cin 3898  ∅c0 4279  ∪ ciun 4951   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ran crn 5652   ∘ ccom 5655  Oncon0 6362   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  ωcom 7877   ↑_m cmap 8847   ≼ cdom 8971  cardccrd 10016  ℕcn 12335   ↾_t crest 17591  SAlgcsalg 47317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-inf2 9642  ax-cc 10513  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-card 10020  df-acn 10023  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-n0 12607  df-z 12694  df-uz 12966  df-rest 17593  df-salg 47318

This theorem is used by:  subsalsal  47368