Theorem smflimlem6 39662

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:   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		smflimlem6.2	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		fvex 6113	. . . . . . . 8 ⊢ (ℤ≥‘𝑀) ∈ V
3	1, 2	eqeltri 2684	. . . . . . 7 ⊢ 𝑍 ∈ V
4		nnex 10903	. . . . . . 7 ⊢ ℕ ∈ V
5	3, 4	xpex 6860	. . . . . 6 ⊢ (𝑍 × ℕ) ∈ V
6	5	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑍 × ℕ) ∈ V)
7		eqid 2610	. . . . . . . . 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 4741	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
10	9	adantr 480	. . . . . . 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 2954	. . . . . 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 7127	. . . . . 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 9239	. . . . 5 ⊢ ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
16	6, 14, 15	sylc 63	. . . 4 ⊢ (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
17	1	uzct 38257	. . . . . . 7 ⊢ 𝑍 ≼ ω
18		nnct 12642	. . . . . . 7 ⊢ ℕ ≼ ω
19	17, 18	pm3.2i 470	. . . . . 6 ⊢ (𝑍 ≼ ω ∧ ℕ ≼ ω)
20		xpct 8722	. . . . . 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 7895	. . . 4 ⊢ ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
24	16, 22, 23	syl2anc 691	. . 3 ⊢ (𝜑 → ran 𝑃 ≼ ω)
25		vex 3176	. . . . . . 7 ⊢ 𝑦 ∈ V
26	12	elrnmpt2g 6670	. . . . . . 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 205	. . . . 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 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))})
30		simp3 1056	. . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
32		smflimlem6.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
33	32	ffvelrnda 6267	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
34	33	adantrr 749	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
35		eqid 2610	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
36		smflimlem6.7	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
38		nnrecre 10934	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
39	38	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
40	37, 39	readdcld 9948	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
41	40	adantrl 748	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
42	31, 34, 35, 41	smfpreimalt 39617	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
43		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
44	43	dmex 6991	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
45	44	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
46		elrest 15911	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)))
50		rabn0 3912	. . . . . . . . . 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 223	. . . . . . . . 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 1074	. . . . . . . 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 2849	. . . . . . 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 1256	. . . . . 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 3019	. . . . 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 480	. . . 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 38425	. 2 ⊢ (𝜑 → ∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦)
59		smflimlem6.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
60	59	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
61	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
62	32	adantr 480	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
66		oveq1 6556	. . . . . . 7 ⊢ (𝑙 = 𝑚 → (𝑙𝑃𝑗) = (𝑚𝑃𝑗))
67	66	fveq2d 6107	. . . . . 6 ⊢ (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
68		oveq2 6557	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
69	68	fveq2d 6107	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
70	67, 69	cbvmpt2v 6633	. . . . 5 ⊢ (𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
71		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑘∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
72		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑗𝑍
73		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑗(ℤ≥‘𝑛)
74		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑚
75		nfmpt22 6621	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
76		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑘
77	74, 75, 76	nfov 6575	. . . . . . . 8 ⊢ Ⅎ𝑗(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78	73, 77	nfiin 4485	. . . . . . 7 ⊢ Ⅎ𝑗∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
79	72, 78	nfiun 4484	. . . . . 6 ⊢ Ⅎ𝑗∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
80		oveq2 6557	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
81	80	adantr 480	. . . . . . . . . 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 4479	. . . . . . . . 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 6556	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
84	83	cbviinv 4496	. . . . . . . . . 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 2644	. . . . . . . 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 480	. . . . . . 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 4478	. . . . . 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 4494	. . . . 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 6103	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑐‘𝑦) = (𝑐‘𝑟))
91		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → 𝑦 = 𝑟)
92	90, 91	eleq12d 2682	. . . . . . 7 ⊢ (𝑦 = 𝑟 → ((𝑐‘𝑦) ∈ 𝑦 ↔ (𝑐‘𝑟) ∈ 𝑟))
93	92	rspccva 3281	. . . . . 6 ⊢ ((∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
94	93	adantll 746	. . . . 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 39661	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))
96	95	ex 449	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
97	96	exlimdv 1848	. 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539  ∅c0 3874  ∪ ciun 4455  ∩ ciin 4456   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ωcom 6957   ≼ cdom 7839  ℝcr 9814  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕcn 10897  ℤcz 11254  ℤ_≥cuz 11563   ⇝ cli 14063   ↾_t crest 15904  SAlgcsalg 39204  SMblFncsmblfn 39586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-acn 8651  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ioo 12050  df-ico 12052  df-fl 12455  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-rest 15906  df-salg 39205  df-smblfn 39587

This theorem is referenced by:  smflim  39663