Theorem smflimlem6 44311

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	1	fvexi 6788	. . . . . . 7 ⊢ 𝑍 ∈ V
3		nnex 11979	. . . . . . 7 ⊢ ℕ ∈ V
4	2, 3	xpex 7603	. . . . . 6 ⊢ (𝑍 × ℕ) ∈ V
5	4	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑍 × ℕ) ∈ V)
6		eqid 2738	. . . . . . . . 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))}
7		smflimlem6.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ SAlg)
8	6, 7	rabexd 5257	. . . . . . . 8 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
9	8	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)
10	9	ralrimivva 3123	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
11		smflimlem6.8	. . . . . . 7 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
12	11	fnmpo 7909	. . . . . 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))
13	10, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 Fn (𝑍 × ℕ))
14		fnrndomg 10292	. . . . 5 ⊢ ((𝑍 × ℕ) ∈ V → (𝑃 Fn (𝑍 × ℕ) → ran 𝑃 ≼ (𝑍 × ℕ)))
15	5, 13, 14	sylc 65	. . . 4 ⊢ (𝜑 → ran 𝑃 ≼ (𝑍 × ℕ))
16	1	uzct 42611	. . . . . . 7 ⊢ 𝑍 ≼ ω
17		nnct 13701	. . . . . . 7 ⊢ ℕ ≼ ω
18	16, 17	pm3.2i 471	. . . . . 6 ⊢ (𝑍 ≼ ω ∧ ℕ ≼ ω)
19		xpct 9772	. . . . . 6 ⊢ ((𝑍 ≼ ω ∧ ℕ ≼ ω) → (𝑍 × ℕ) ≼ ω)
20	18, 19	ax-mp 5	. . . . 5 ⊢ (𝑍 × ℕ) ≼ ω
21	20	a1i 11	. . . 4 ⊢ (𝜑 → (𝑍 × ℕ) ≼ ω)
22		domtr 8793	. . . 4 ⊢ ((ran 𝑃 ≼ (𝑍 × ℕ) ∧ (𝑍 × ℕ) ≼ ω) → ran 𝑃 ≼ ω)
23	15, 21, 22	syl2anc 584	. . 3 ⊢ (𝜑 → ran 𝑃 ≼ ω)
24		vex 3436	. . . . . . 7 ⊢ 𝑦 ∈ V
25	11	elrnmpog 7409	. . . . . . 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))}))
26	24, 25	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))})
27	26	biimpi 215	. . . . 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))})
28	27	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))})
29		simp3 1137	. . . . . . . 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))})
30	7	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → 𝑆 ∈ SAlg)
31		smflimlem6.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
32	31	ffvelrnda 6961	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
33	32	adantrr 714	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
34		eqid 2738	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
35		smflimlem6.7	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ ℝ)
36	35	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ ℝ)
37		nnrecre 12015	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ)
38	37	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (1 / 𝑘) ∈ ℝ)
39	36, 38	readdcld 11004	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
40	39	adantrl 713	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → (𝐴 + (1 / 𝑘)) ∈ ℝ)
41	30, 33, 34, 40	smfpreimalt 44267	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∈ (𝑆 ↾t dom (𝐹‘𝑚)))
42		fvex 6787	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
43	42	dmex 7758	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
44	43	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → dom (𝐹‘𝑚) ∈ V)
45		elrest 17138	. . . . . . . . . . . . 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))))
46	7, 44, 45	syl2anc 584	. . . . . . . . . . . 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))))
47	46	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))))
48	41, 47	mpbid 231	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → ∃𝑠 ∈ 𝑆 {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)))
49		rabn0 4319	. . . . . . . . . 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)))
50	48, 49	sylibr 233	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ)) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ≠ ∅)
51	50	3adant3 1131	. . . . . . . 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))} ≠ ∅)
52	29, 51	eqnetrd 3011	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ) ∧ 𝑦 = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) → 𝑦 ≠ ∅)
53	52	3exp 1118	. . . . . 6 ⊢ (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	rexlimdvv 3222	. . . . 5 ⊢ (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	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 ≠ ∅))
56	28, 55	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → 𝑦 ≠ ∅)
57	23, 56	axccd2 42769	. 2 ⊢ (𝜑 → ∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦)
58		smflimlem6.1	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
59	58	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑀 ∈ ℤ)
60	7	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
61	31	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐹:𝑍⟶(SMblFn‘𝑆))
62		smflimlem6.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
63		smflimlem6.6	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
64	35	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → 𝐴 ∈ ℝ)
65		fvoveq1 7298	. . . . . 6 ⊢ (𝑙 = 𝑚 → (𝑐‘(𝑙𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑗)))
66		oveq2 7283	. . . . . . 7 ⊢ (𝑗 = 𝑘 → (𝑚𝑃𝑗) = (𝑚𝑃𝑘))
67	66	fveq2d 6778	. . . . . 6 ⊢ (𝑗 = 𝑘 → (𝑐‘(𝑚𝑃𝑗)) = (𝑐‘(𝑚𝑃𝑘)))
68	65, 67	cbvmpov 7370	. . . . 5 ⊢ (𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗))) = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝑐‘(𝑚𝑃𝑘)))
69		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑘∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗)
70		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑗𝑍
71		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑗(ℤ≥‘𝑛)
72		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑚
73		nfmpo2 7356	. . . . . . . . 9 ⊢ Ⅎ𝑗(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))
74		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑗𝑘
75	72, 73, 74	nfov 7305	. . . . . . . 8 ⊢ Ⅎ𝑗(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
76	71, 75	nfiin 4955	. . . . . . 7 ⊢ Ⅎ𝑗∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
77	70, 76	nfiun 4954	. . . . . 6 ⊢ Ⅎ𝑗∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
78		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
79	78	adantr 481	. . . . . . . . . 10 ⊢ ((𝑗 = 𝑘 ∧ 𝑖 ∈ (ℤ≥‘𝑛)) → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑗) = (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
80	79	iineq2dv 4949	. . . . . . . . 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))
81		oveq1 7282	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑚 → (𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = (𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘))
82	81	cbviinv 4971	. . . . . . . . . 10 ⊢ ∩ 𝑖 ∈ (ℤ≥‘𝑛)(𝑖(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)(𝑚(𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ (𝑐‘(𝑙𝑃𝑗)))𝑘)
83	82	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))
84	80, 83	eqtrd 2778	. . . . . . . 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))
85	84	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))
86	85	iuneq2dv 4948	. . . . . 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))
87	69, 77, 86	cbviin 4967	. . . . 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)
88		fveq2 6774	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → (𝑐‘𝑦) = (𝑐‘𝑟))
89		id 22	. . . . . . . 8 ⊢ (𝑦 = 𝑟 → 𝑦 = 𝑟)
90	88, 89	eleq12d 2833	. . . . . . 7 ⊢ (𝑦 = 𝑟 → ((𝑐‘𝑦) ∈ 𝑦 ↔ (𝑐‘𝑟) ∈ 𝑟))
91	90	rspccva 3560	. . . . . 6 ⊢ ((∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
92	91	adantll 711	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) ∧ 𝑟 ∈ ran 𝑃) → (𝑐‘𝑟) ∈ 𝑟)
93	59, 1, 60, 61, 62, 63, 64, 11, 68, 87, 92	smflimlem5 44310	. . . 4 ⊢ ((𝜑 ∧ ∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦) → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))
94	93	ex 413	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
95	94	exlimdv 1936	. 2 ⊢ (𝜑 → (∃𝑐∀𝑦 ∈ ran 𝑃(𝑐‘𝑦) ∈ 𝑦 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷)))
96	57, 95	mpd 15	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ∈ (𝑆 ↾t 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432   ∩ cin 3886  ∅c0 4256  ∪ ciun 4924  ∩ ciin 4925   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  dom cdm 5589  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  ωcom 7712   ≼ cdom 8731  ℝcr 10870  1c1 10872   + caddc 10874   < clt 11009   ≤ cle 11010   / cdiv 11632  ℕcn 11973  ℤcz 12319  ℤ_≥cuz 12582   ⇝ cli 15193   ↾_t crest 17131  SAlgcsalg 43849  SMblFncsmblfn 44233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cc 10191  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-map 8617  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-oi 9269  df-card 9697  df-acn 9700  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12583  df-q 12689  df-rp 12731  df-ioo 13083  df-ico 13085  df-fl 13512  df-seq 13722  df-exp 13783  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-rlim 15198  df-rest 17133  df-salg 43850  df-smblfn 44234

This theorem is referenced by:  smflim  44312