Theorem smflimsuplem1 46775

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 6906	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
3	2	fveq1d 6908	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑗)‘𝑥))
4	3	cbvmptv 5260	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥))
5	4	rneqi 5950	. . . . . . . . 9 ⊢ ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥))
6	5	supeq1i 9484	. . . . . . . 8 ⊢ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )
7	6	mpteq2i 5252	. . . . . . 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 6906	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (𝐸‘𝑛) = (𝐸‘𝐾))
10		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑛 = 𝐾 → (ℤ≥‘𝑛) = (ℤ≥‘𝐾))
11	10	mpteq1d 5242	. . . . . . . . 9 ⊢ (𝑛 = 𝐾 → (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)) = (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)))
12	11	rneqd 5951	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → ran (𝑗 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑗)‘𝑥)) = ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)))
13	12	supeq1d 9483	. . . . . . 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 5238	. . . . . 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 2774	. . . . 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 6919	. . . . . . 7 ⊢ (𝐸‘𝐾) ∈ V
18	17	mptex 7242	. . . . . 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 7038	. . . 4 ⊢ (𝜑 → (𝐻‘𝐾) = (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
21	20	dmeqd 5918	. . 3 ⊢ (𝜑 → dom (𝐻‘𝐾) = dom (𝑥 ∈ (𝐸‘𝐾) ↦ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < )))
22		xrltso 13179	. . . . . 6 ⊢ < Or ℝ*
23	22	supex 9500	. . . . 5 ⊢ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ V
24		eqid 2734	. . . . 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 6712	. . . 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 5918	. . . . . . . . . 10 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
29	28	cbviinv 5045	. . . . . . . . 9 ⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗)
30	29	eleq2i 2830	. . . . . . . 8 ⊢ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗))
31	6	eleq1i 2829	. . . . . . . 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 3435	. . . . . 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 45029	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) = ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗))
36	35	eleq2d 2824	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑗) ↔ 𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗)))
37	13	eleq1d 2823	. . . . . . 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 3434	. . . . 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 2774	. . . 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 2734	. . . . 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 45362	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℤ)
44		uzid 12890	. . . . . . 7 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ≥‘𝐾))
45		ne0i 4346	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝐾) ≠ ∅)
46	43, 44, 45	3syl 18	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝐾) ≠ ∅)
47		fvex 6919	. . . . . . . . 9 ⊢ (𝐹‘𝑗) ∈ V
48	47	dmex 7931	. . . . . . . 8 ⊢ dom (𝐹‘𝑗) ∈ V
49	48	rgenw 3062	. . . . . . 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 45072	. . . . 5 ⊢ (𝜑 → ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∈ V)
52	41, 51	rabexd 5345	. . . 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 7038	. . 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 2778	. 2 ⊢ (𝜑 → dom (𝐻‘𝐾) = {𝑥 ∈ ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ∣ sup(ran (𝑗 ∈ (ℤ≥‘𝐾) ↦ ((𝐹‘𝑗)‘𝑥)), ℝ*, < ) ∈ ℝ})
55		ssrab2 4089	. . . 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 6906	. . . . 5 ⊢ (𝑗 = 𝐾 → (𝐹‘𝑗) = (𝐹‘𝐾))
59	58	dmeqd 5918	. . . 4 ⊢ (𝑗 = 𝐾 → dom (𝐹‘𝑗) = dom (𝐹‘𝐾))
60		ssid 4017	. . . . 5 ⊢ dom (𝐹‘𝐾) ⊆ dom (𝐹‘𝐾)
61	60	a1i 11	. . . 4 ⊢ (𝜑 → dom (𝐹‘𝐾) ⊆ dom (𝐹‘𝐾))
62	57, 59, 61	iinssd 45070	. . 3 ⊢ (𝜑 → ∩ 𝑗 ∈ (ℤ≥‘𝐾)dom (𝐹‘𝑗) ⊆ dom (𝐹‘𝐾))
63	56, 62	sstrd 4005	. 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 4033	1 ⊢ (𝜑 → dom (𝐻‘𝐾) ⊆ dom (𝐹‘𝐾))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  {crab 3432  Vcvv 3477   ⊆ wss 3962  ∅c0 4338  ∩ ciin 4996   ↦ cmpt 5230  dom cdm 5688  ran crn 5689  ‘cfv 6562  supcsup 9477  ℝcr 11151  ℝ^*cxr 11291   < clt 11292  ℤcz 12610  ℤ_≥cuz 12875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-pre-lttri 11226  ax-pre-lttrn 11227

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-sup 9479  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-neg 11492  df-z 12611  df-uz 12876

This theorem is referenced by:  smflimsuplem4  46778