Theorem smflimsuplem1 46857

Description: If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflimsuplem1.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smflimsuplem1.e	⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem1.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem1.k	⊢ (𝜑 → 𝐾 ∈ 𝑍)

Assertion

Ref	Expression
smflimsuplem1	⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾))

Distinct variable groups:   𝑛,𝐸,𝑥   𝑚,𝐹,𝑛,𝑥   𝑛,𝐾,𝑥   𝑛,𝑍

Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝐸(𝑚)   𝐻(𝑥,𝑚,𝑛)   𝐾(𝑚)   𝑀(𝑥,𝑚,𝑛)   𝑍(𝑥,𝑚)

Proof of Theorem smflimsuplem1

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsuplem1.h	. . . . 5 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
2		fveq2 6822	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
3	2	fveq1d 6824	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑗)‘𝑥))
4	3	cbvmptv 5195	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥))
5	4	rneqi 5877	. . . . . . . . 9 ⊢ ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥))
6	5	supeq1i 9331	. . . . . . . 8 ⊢ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )
7	6	mpteq2i 5187	. . . . . . 7 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ))
8	7	a1i 11	. . . . . 6 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
9		fveq2 6822	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
10		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑛 = 𝐾 → (ℤ≥‘𝑛) = (ℤ≥‘𝐾))
11	10	mpteq1d 5181	. . . . . . . . 9 ⊢ (𝑛 = 𝐾 → (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)))
12	11	rneqd 5878	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)))
13	12	supeq1d 9330	. . . . . . 7 ⊢ (𝑛 = 𝐾 → sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ))
14	9, 13	mpteq12dv 5178	. . . . . 6 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
15	8, 14	eqtrd 2766	. . . . 5 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
16		smflimsuplem1.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
17		fvex 6835	. . . . . . 7 ⊢ (𝐸‘𝐾) ∈ V
18	17	mptex 7157	. . . . . 6 ⊢ (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )) ∈ V
19	18	a1i 11	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )) ∈ V)
20	1, 15, 16, 19	fvmptd3 6952	. . . 4 ⊢ (𝜑 → (𝐻‘𝐾) = (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
21	20	dmeqd 5845	. . 3 ⊢ (𝜑 → dom (𝐻‘𝐾) = dom (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
22		xrltso 13037	. . . . . 6 ⊢ < Or ℝ*
23	22	supex 9348	. . . . 5 ⊢ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ V
24		eqid 2731	. . . . 5 ⊢ (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )) = (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ))
25	23, 24	dmmpti 6625	. . . 4 ⊢ dom (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )) = (𝐸‘𝐾)
26	25	a1i 11	. . 3 ⊢ (𝜑 → dom (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )) = (𝐸‘𝐾))
27		smflimsuplem1.e	. . . 4 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
28	2	dmeqd 5845	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
29	28	cbviinv 4990	. . . . . . . . 9 ⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗)
30	29	eleq2i 2823	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗))
31	6	eleq1i 2822	. . . . . . . 8 ⊢ (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)
32	30, 31	anbi12i 628	. . . . . . 7 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ∧ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
33	32	rabbia2 3398	. . . . . 6 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ}
34	33	a1i 11	. . . . 5 ⊢ (𝑛 = 𝐾 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
35	10	iineq1d 45126	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) = ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗))
36	35	eleq2d 2817	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ↔ 𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗)))
37	13	eleq1d 2816	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ))
38	36, 37	anbi12d 632	. . . . . 6 ⊢ (𝑛 = 𝐾 → ((𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ∧ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∧ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ)))
39	38	rabbidva2 3397	. . . . 5 ⊢ (𝑛 = 𝐾 → {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
40	34, 39	eqtrd 2766	. . . 4 ⊢ (𝑛 = 𝐾 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
41		eqid 2731	. . . . 5 ⊢ {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ}
42		smflimsuplem1.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ≥‘𝑀)
43	42, 16	eluzelz2d 45450	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ)
44		uzid 12744	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ≥‘𝐾))
45		ne0i 4291	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝐾) ≠ ∅)
46	43, 44, 45	3syl 18	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝐾) ≠ ∅)
47		fvex 6835	. . . . . . . . 9 ⊢ (𝐹‘𝑗) ∈ V
48	47	dmex 7839	. . . . . . . 8 ⊢ dom (𝐹‘𝑗) ∈ V
49	48	rgenw 3051	. . . . . . 7 ⊢ ∀𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∈ V
50	49	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∈ V)
51	46, 50	iinexd 45169	. . . . 5 ⊢ (𝜑 → ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∈ V)
52	41, 51	rabexd 5278	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
53	27, 40, 16, 52	fvmptd3 6952	. . 3 ⊢ (𝜑 → (𝐸‘𝐾) = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
54	21, 26, 53	3eqtrd 2770	. 2 ⊢ (𝜑 → dom (𝐻‘𝐾) = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55		ssrab2 4030	. . . 4 ⊢ {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗)
56	55	a1i 11	. . 3 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗))
57	43, 44	syl 17	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝐾))
58		fveq2 6822	. . . . 5 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
59	58	dmeqd 5845	. . . 4 ⊢ (𝑗 = 𝐾 → dom (𝐹‘𝑗) = dom (𝐹‘𝐾))
60		ssid 3957	. . . . 5 ⊢ dom (𝐹‘𝐾) ⊆ dom (𝐹‘𝐾)
61	60	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐹‘𝐾) ⊆ dom (𝐹‘𝐾))
62	57, 59, 61	iinssd 45167	. . 3 ⊢ (𝜑 → ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ⊆ dom (𝐹‘𝐾))
63	56, 62	sstrd 3945	. 2 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ} ⊆ dom (𝐹‘𝐾))
64	54, 63	eqsstrd 3969	1 ⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  {crab 3395  Vcvv 3436   ⊆ wss 3902  ∅c0 4283  ∩ ciin 4942   ↦ cmpt 5172  dom cdm 5616  ran crn 5617  ‘cfv 6481  supcsup 9324  ℝcr 11002  ℝ^*cxr 11142   < clt 11143  ℤcz 12465  ℤ_≥cuz 12729

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-pre-lttri 11077  ax-pre-lttrn 11078

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-iin 4944  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-po 5524  df-so 5525  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-neg 11344  df-z 12466  df-uz 12730

This theorem is referenced by:  smflimsuplem4  46860