Theorem smflimsuplem1 47392

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 6871	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
3	2	fveq1d 6873	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑗)‘𝑥))
4	3	cbvmptv 5209	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥))
5	4	rneqi 5918	. . . . . . . . 9 ⊢ ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥))
6	5	supeq1i 9395	. . . . . . . 8 ⊢ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )
7	6	mpteq2i 5201	. . . . . . 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 6871	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
10		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑛 = 𝐾 → (ℤ≥‘𝑛) = (ℤ≥‘𝐾))
11	10	mpteq1d 5195	. . . . . . . . 9 ⊢ (𝑛 = 𝐾 → (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)))
12	11	rneqd 5919	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)))
13	12	supeq1d 9394	. . . . . . 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 5192	. . . . . 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 2800	. . . . 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 6884	. . . . . . 7 ⊢ (𝐸‘𝐾) ∈ V
18	17	mptex 7211	. . . . . 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 7003	. . . 4 ⊢ (𝜑 → (𝐻‘𝐾) = (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
21	20	dmeqd 5886	. . 3 ⊢ (𝜑 → dom (𝐻‘𝐾) = dom (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
22		xrltso 13157	. . . . . 6 ⊢ < Or ℝ*
23	22	supex 9412	. . . . 5 ⊢ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ V
24		eqid 2765	. . . . 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 6669	. . . 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 5886	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
29	28	cbviinv 5000	. . . . . . . . 9 ⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗)
30	29	eleq2i 2857	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗))
31	6	eleq1i 2856	. . . . . . . 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 639	. . . . . . 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 3420	. . . . . 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 45666	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) = ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗))
36	35	eleq2d 2851	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ↔ 𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗)))
37	13	eleq1d 2850	. . . . . . 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 643	. . . . . 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 3419	. . . . 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 2800	. . . 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 2765	. . . . 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 45985	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ)
44		uzid 12868	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ≥‘𝐾))
45		ne0i 4296	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝐾) ≠ ∅)
46	43, 44, 45	3syl 19	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝐾) ≠ ∅)
47		fvex 6884	. . . . . . . . 9 ⊢ (𝐹‘𝑗) ∈ V
48	47	dmex 7894	. . . . . . . 8 ⊢ dom (𝐹‘𝑗) ∈ V
49	48	rgenw 3083	. . . . . . 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 45709	. . . . 5 ⊢ (𝜑 → ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∈ V)
52	41, 51	rabexd 5301	. . . 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 7003	. . 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 2804	. 2 ⊢ (𝜑 → dom (𝐻‘𝐾) = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55		ssrab2 4036	. . . 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 18	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝐾))
58		fveq2 6871	. . . . 5 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
59	58	dmeqd 5886	. . . 4 ⊢ (𝑗 = 𝐾 → dom (𝐹‘𝑗) = dom (𝐹‘𝐾))
60		ssid 3961	. . . . 5 ⊢ dom (𝐹‘𝐾) ⊆ dom (𝐹‘𝐾)
61	60	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐹‘𝐾) ⊆ dom (𝐹‘𝐾))
62	57, 59, 61	iinssd 45707	. . 3 ⊢ (𝜑 → ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ⊆ dom (𝐹‘𝐾))
63	56, 62	sstrd 3949	. 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 3973	1 ⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  ∩ ciin 4953   ↦ cmpt 5186  dom cdm 5652  ran crn 5653  ‘cfv 6525  supcsup 9388  ℝcr 11087  ℝ^*cxr 11230   < clt 11231  ℤcz 12582  ℤ_≥cuz 12853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-pre-lttri 11162  ax-pre-lttrn 11163

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-po 5560  df-so 5561  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-er 8682  df-en 8932  df-dom 8933  df-sdom 8934  df-sup 9390  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-neg 11432  df-z 12583  df-uz 12854

This theorem is referenced by:  smflimsuplem4  47395