Theorem smflimlem6 40984

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. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

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:   𝑥,𝑘,𝐴,𝑚,𝑛   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		smflimlem6.2	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		fvex 6201	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ∈ V
3	1, 2	eqeltri 2697	. . . . . . 7 ⊢ 𝑍 ∈ V
4		nnex 11026	. . . . . . 7 ⊢ ℕ ∈ V
5	3, 4	xpex 6962	. . . . . 6 ⊢ (𝑍 × ℕ) ∈ V
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑍 × ℕ) ∈ V)
7		eqid 2622	. . . . . . . . 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))}
8		smflimlem6.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
9	7, 8	rabexd 4814	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
10	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
11	10	ralrimivva 2971	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
12		smflimlem6.8	. . . . . . 7 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
13	12	fnmpt2 7238	. . . . . 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))
14	11, 13	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 Fn (𝑍 × ℕ))
15		fnrndomg 9358	. . . . 5 ⊢ ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
16	6, 14, 15	sylc 65	. . . 4 ⊢ (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
17	1	uzct 39232	. . . . . . 7 ⊢ 𝑍 ≼ ω
18		nnct 12780	. . . . . . 7 ⊢ ℕ ≼ ω
19	17, 18	pm3.2i 471	. . . . . 6 ⊢ (𝑍 ≼ ω ∧ ℕ ≼ ω)
20		xpct 8839	. . . . . 6 ⊢ ((𝑍 ≼ ω ∧ ℕ ≼ ω) → (𝑍 × ℕ) ≼ ω)
21	19, 20	ax-mp 5	. . . . 5 ⊢ (𝑍 × ℕ) ≼ ω
22	21	a1i 11	. . . 4 ⊢ (𝜑 → (𝑍 × ℕ) ≼ ω)
23		domtr 8009	. . . 4 ⊢ ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
24	16, 22, 23	syl2anc 693	. . 3 ⊢ (𝜑 → ran 𝑃 ≼ ω)
25		vex 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
26	12	elrnmpt2g 6772	. . . . . . 7 ⊢ (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))}))
27	25, 26	ax-mp 5	. . . . . 6 ⊢ (𝑦 ∈ ran 𝑃 ↔ ∃𝑚 ∈ 𝑍 ∃𝑘 ∈ ℕ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
28	27	biimpi 206	. . . . 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))})
29	28	adantl 482	. . . 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))})
30		simp3 1063	. . . . . . . 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))})
31	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
32		smflimlem6.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
33	32	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
34	33	adantrr 753	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
35		eqid 2622	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
36		smflimlem6.7	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
37	36	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38		nnrecre 11057	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
39	38	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
40	37, 39	readdcld 10069	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
41	40	adantrl 752	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
42	31, 34, 35, 41	smfpreimalt 40940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
43		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
44	43	dmex 7099	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
46		elrest 16088	. . . . . . . . . . . . 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))))
47	8, 45, 46	syl2anc 693	. . . . . . . . . . . 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))))
48	47	adantr 481	. . . . . . . . . . 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))))
49	42, 48	mpbid 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)))
50		rabn0 3958	. . . . . . . . . 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)))
51	49, 50	sylibr 224	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅)
52	51	3adant3 1081	. . . . . . . 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))} ≠ ∅)
53	30, 52	eqnetrd 2861	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 ≠ ∅)
54	53	3exp 1264	. . . . . 6 ⊢ (𝜑 → ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) → (𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} → 𝑦 ≠ ∅)))
55	54	rexlimdvv 3037	. . . . 5 ⊢ (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	adantr 481	. . . 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 ≠ ∅))
57	29, 56	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → 𝑦 ≠ ∅)
58	24, 57	axccd2 39430	. 2 ⊢ (𝜑 → ∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦)
59		smflimlem6.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60	59	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
61	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
62	32	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐹:𝑍⟶(SMblFn‘𝑆))
63		smflimlem6.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
64		smflimlem6.6	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
65	36	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
66		oveq1 6657	. . . . . . 7 ⊢ (𝑙 = 𝑚 → (𝑙𝑃𝑗) = (𝑚𝑃𝑗))
67	66	fveq2d 6195	. . . . . 6 ⊢ (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
68		oveq2 6658	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
69	68	fveq2d 6195	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
70	67, 69	cbvmpt2v 6735	. . . . 5 ⊢ (𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
71		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑘∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
72		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑗𝑍
73		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑗(ℤ≥‘𝑛)
74		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑚
75		nfmpt22 6723	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
76		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑘
77	74, 75, 76	nfov 6676	. . . . . . . 8 ⊢ Ⅎ𝑗(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78	73, 77	nfiin 4549	. . . . . . 7 ⊢ Ⅎ𝑗∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
79	72, 78	nfiun 4548	. . . . . 6 ⊢ Ⅎ𝑗∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
80		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
81	80	adantr 481	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 ∈ (ℤ≥‘𝑛)) → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
82	81	iineq2dv 4543	. . . . . . . . 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))
83		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
84	83	cbviinv 4560	. . . . . . . . . 10 ⊢ ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
85	84	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))
86	82, 85	eqtrd 2656	. . . . . . . 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))
87	86	adantr 481	. . . . . . 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))
88	87	iuneq2dv 4542	. . . . . 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))
89	71, 79, 88	cbviin 4558	. . . . 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)
90		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑐‘𝑦) = (𝑐‘𝑟))
91		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → 𝑦 = 𝑟)
92	90, 91	eleq12d 2695	. . . . . . 7 ⊢ (𝑦 = 𝑟 → ((𝑐‘𝑦) ∈ 𝑦 ↔ (𝑐‘𝑟) ∈ 𝑟))
93	92	rspccva 3308	. . . . . 6 ⊢ ((∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
94	93	adantll 750	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
95	60, 1, 61, 62, 63, 64, 65, 12, 70, 89, 94	smflimlem5 40983	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))
96	95	ex 450	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
97	96	exlimdv 1861	. 2 ⊢ (𝜑 → (∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
98	58, 97	mpd 15	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))

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  ∩ ciin 4521   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065   ≼ cdom 7953  ℝcr 9935  1c1 9937   + caddc 9939   < clt 10074   ≤ cle 10075   / cdiv 10684  ℕcn 11020  ℤcz 11377  ℤ_≥cuz 11687   ⇝ cli 14215   ↾_t crest 16081  SAlgcsalg 40528  SMblFncsmblfn 40909

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  ax-pre-sup 10014

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-iin 4523  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-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smflim  40985