Theorem smflimlem6 47880

Description: Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. The proof uses fnrndomnum 10621 rather than fnrndomg 10622, and so does not require ax-ac 10541. (Contributed by Glauco Siliprandi, 26-Jun-2021.) (Revised by Vincent Gonzalez, 30-Aug-2026.)

Hypotheses

Ref	Expression
smflimlem6.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimlem6.2	⊢ 𝑍 = (ℤ≥‘𝑀)
smflimlem6.3	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimlem6.4	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimlem6.5	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem6.6	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
smflimlem6.7	⊢ (𝜑 → 𝐴 ∈ ℝ)
smflimlem6.8	⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})

Assertion

Ref	Expression
smflimlem6	⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))

Distinct variable groups:   𝐴,𝑘,𝑚,𝑛,𝑥   𝐴,𝑠,𝑘,𝑚,𝑥   𝐷,𝑘,𝑚,𝑛,𝑥   𝑘,𝐹,𝑚,𝑛,𝑥   𝐹,𝑠   𝑘,𝐺,𝑚,𝑛   𝑚,𝑀   𝑃,𝑘,𝑚,𝑛,𝑥   𝑃,𝑠   𝑆,𝑘,𝑚,𝑛   𝑆,𝑠   𝑘,𝑍,𝑚,𝑛,𝑥   𝑍,𝑠   𝜑,𝑘,𝑚,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑠)   𝐷(𝑠)   𝑆(𝑥)   𝐺(𝑥, 𝑠)   𝑀(𝑥, 𝑘, 𝑛, 𝑠)

Proof of Theorem smflimlem6

Dummy variables 𝑐 𝑟 𝑖 𝑗 𝑙 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9651	. . . . . . . 8 ⊢ ω ∈ On
2		smflimlem6.2	. . . . . . . . 9 ⊢ 𝑍 = (ℤ≥‘𝑀)
3	2	uzct 46174	. . . . . . . 8 ⊢ 𝑍 ≼ ω
4		ondomen 10120	. . . . . . . 8 ⊢ ((ω ∈ On ∧ 𝑍 ≼ ω) → 𝑍 ∈ dom card)
5	1, 3, 4	mp2an 705	. . . . . . 7 ⊢ 𝑍 ∈ dom card
6		nnct 14128	. . . . . . . 8 ⊢ ℕ ≼ ω
7		ondomen 10120	. . . . . . . 8 ⊢ ((ω ∈ On ∧ ℕ ≼ ω) → ℕ ∈ dom card)
8	1, 6, 7	mp2an 705	. . . . . . 7 ⊢ ℕ ∈ dom card
9		xpnum 10036	. . . . . . 7 ⊢ ((𝑍 ∈ dom card ∧ ℕ ∈ dom card) → (𝑍 × ℕ) ∈ dom card)
10	5, 8, 9	mp2an 705	. . . . . 6 ⊢ (𝑍 × ℕ) ∈ dom card
11	10	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑍 × ℕ) ∈ dom card)
12		eqid 2760	. . . . . . . . 9 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
13		smflimlem6.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
14	12, 13	rabexd 5300	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
15	14	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
16	15	ralrimivva 3205	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
17		smflimlem6.8	. . . . . . 7 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
18	17	fnmpo 8081	. . . . . 6 ⊢ (∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V → 𝑃 Fn (𝑍 × ℕ))
19	16, 18	syl 18	. . . . 5 ⊢ (𝜑 → 𝑃 Fn (𝑍 × ℕ))
20		fnrndomnum 10621	. . . . 5 ⊢ ((𝑍 × ℕ) ∈ dom card → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
21	11, 19, 20	sylc 66	. . . 4 ⊢ (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
22	3, 6	pm3.2i 476	. . . . . 6 ⊢ (𝑍 ≼ ω ∧ ℕ ≼ ω)
23		xpct 10099	. . . . . 6 ⊢ ((𝑍 ≼ ω ∧ ℕ ≼ ω) → (𝑍 × ℕ) ≼ ω)
24	22, 23	ax-mp 5	. . . . 5 ⊢ (𝑍 × ℕ) ≼ ω
25	24	a1i 11	. . . 4 ⊢ (𝜑 → (𝑍 × ℕ) ≼ ω)
26		domtr 9038	. . . 4 ⊢ ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
27	21, 25, 26	syl2anc 596	. . 3 ⊢ (𝜑 → ran 𝑃 ≼ ω)
28		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
29	17	elrnmpog 7556	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ran 𝑃 ↔ ∃𝑚 ∈ 𝑍 ∃𝑘 ∈ ℕ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}))
30	28, 29	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ ran 𝑃 ↔ ∃𝑚 ∈ 𝑍 ∃𝑘 ∈ ℕ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
31	30	bilani 510	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → ∃𝑚 ∈ 𝑍 ∃𝑘 ∈ ℕ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
32		simp3 1156	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
33	13	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
34		smflimlem6.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
35	34	ffvelcdmda 7085	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
36	35	adantrr 730	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
37		eqid 2760	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
38		smflimlem6.7	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	38	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
40		nnrecre 12384	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
41	40	adantl 487	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
42	39, 41	readdcld 11342	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
43	42	adantrl 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
44	33, 36, 37, 43	smfpreimalt 47835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
45		fvex 6899	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
46	45	dmex 7922	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
47	46	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
48		elrest 17604	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ SAlg ∧ dom (𝐹‘𝑚) ∈ V) → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))))
49	13, 47, 48	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))))
50	49	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)) ↔ ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))))
51	44, 50	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)))
52		rabn0 4338	. . . . . . . . . 10 ⊢ ({𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅ ↔ ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)))
53	51, 52	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅)
54	53	3adant3 1150	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅)
55	32, 54	eqnetrd 3022	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 ≠ ∅)
56	55	3exp 1137	. . . . . 6 ⊢ (𝜑 → ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) → (𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅)))
57	56	rexlimdvv 3218	. . . . 5 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 ∃𝑘 ∈ ℕ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅))
58	57	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → (∃𝑚 ∈ 𝑍 ∃𝑘 ∈ ℕ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅))
59	31, 58	mpd 16	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → 𝑦 ≠ ∅)
60	27, 59	axccd2 46336	. 2 ⊢ (𝜑 → ∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦)
61		smflimlem6.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
62	61	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
63	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
64	34	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐹:𝑍⟶(SMblFn‘𝑆))
65		smflimlem6.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
66		smflimlem6.6	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
67	38	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
68		fvoveq1 7444	. . . . . 6 ⊢ (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
69		oveq2 7429	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
70	69	fveq2d 6890	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
71	68, 70	cbvmpov 7516	. . . . 5 ⊢ (𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
72		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑘∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
73		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑗𝑍
74		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑗(ℤ≥‘𝑛)
75		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑚
76		nfmpo2 7502	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
77		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑘
78	75, 76, 77	nfov 7451	. . . . . . . 8 ⊢ Ⅎ𝑗(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
79	74, 78	nfiin 4982	. . . . . . 7 ⊢ Ⅎ𝑗∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
80	73, 79	nfiun 4981	. . . . . 6 ⊢ Ⅎ𝑗∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
81		oveq2 7429	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
82	81	adantr 486	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 ∈ (ℤ≥‘𝑛)) → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
83	82	iineq2dv 4976	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
84		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
85	84	cbviinv 4997	. . . . . . . . . 10 ⊢ ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
86	85	a1i 11	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
87	83, 86	eqtrd 2795	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
88	87	adantr 486	. . . . . . 7 ⊢ ((𝑗 = 𝑘 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
89	88	iuneq2dv 4975	. . . . . 6 ⊢ (𝑗 = 𝑘 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
90	72, 80, 89	cbviin 4993	. . . . 5 ⊢ ∩ 𝑗 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
91		fveq2 6886	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑐‘𝑦) = (𝑐‘𝑟))
92		id 23	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → 𝑦 = 𝑟)
93	91, 92	eleq12d 2854	. . . . . . 7 ⊢ (𝑦 = 𝑟 → ((𝑐‘𝑦) ∈ 𝑦 ↔ (𝑐‘𝑟) ∈ 𝑟))
94	93	rspccva 3575	. . . . . 6 ⊢ ((∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
95	94	adantll 727	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
96	62, 2, 63, 64, 65, 66, 67, 17, 71, 90, 95	smflimlem5 47879	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))
97	96	ex 418	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
98	97	exlimdv 1965	. 2 ⊢ (𝜑 → (∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
99	60, 98	mpd 16	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ∩ cin 4089  ∅c0 4278  ∪ ciun 4950  ∩ ciin 4951   class class class wbr 5102   ↦ cmpt 5185   × cxp 5648  dom cdm 5650  ran crn 5651  Oncon0 6362   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  ωcom 7878   ≼ cdom 8975  cardccrd 10020  ℝcr 11203  1c1 11205   + caddc 11207   < clt 11347   ≤ cle 11348   / cdiv 11977  ℕcn 12339  ℤcz 12697  ℤ_≥cuz 12969   ⇝ cli 15657   ↾_t crest 17597  SAlgcsalg 47412  SMblFncsmblfn 47799

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-inf2 9646  ax-cc 10517  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281  ax-pre-sup 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-oadd 8484  df-omul 8485  df-er 8721  df-map 8853  df-pm 8854  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-sup 9438  df-inf 9439  df-oi 9508  df-card 10024  df-acn 10027  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-div 11978  df-nn 12340  df-2 12409  df-3 12410  df-n0 12611  df-z 12698  df-uz 12970  df-q 13080  df-rp 13125  df-ioo 13484  df-ico 13486  df-fl 13936  df-seq 14149  df-exp 14209  df-cj 15272  df-re 15273  df-im 15274  df-sqrt 15408  df-abs 15409  df-clim 15661  df-rlim 15662  df-rest 17599  df-salg 47413  df-smblfn 47800

This theorem is used by:  smflim  47881