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Theorem smfpimcc 43089
Description: Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function that, for each measurable function, chooses a measurable set that, when intersected with the function's domain, gives the preimage of 𝐴. This is a generalization of the observation at the beginning of the proof of Proposition 121F of [Fremlin1] p. 39 . The statement would also be provable for uncountable sets, but in most cases it will suffice to consider the countable case, and only the axiom of countable choice will be needed. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smfpimcc.1 𝑛𝐹
smfpimcc.z 𝑍 = (ℤ𝑀)
smfpimcc.s (𝜑𝑆 ∈ SAlg)
smfpimcc.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smfpimcc.j 𝐽 = (topGen‘ran (,))
smfpimcc.b 𝐵 = (SalGen‘𝐽)
smfpimcc.a (𝜑𝐴𝐵)
Assertion
Ref Expression
smfpimcc (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
Distinct variable groups:   𝐴,,𝑛   ,𝐹   𝑆,   ,𝑍,𝑛
Allowed substitution hints:   𝜑(,𝑛)   𝐵(,𝑛)   𝑆(𝑛)   𝐹(𝑛)   𝐽(,𝑛)   𝑀(,𝑛)

Proof of Theorem smfpimcc
Dummy variables 𝑓 𝑚 𝑠 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smfpimcc.z . . . . . . 7 𝑍 = (ℤ𝑀)
21uzct 41331 . . . . . 6 𝑍 ≼ ω
32a1i 11 . . . . 5 (𝜑𝑍 ≼ ω)
4 mptct 9963 . . . . 5 (𝑍 ≼ ω → (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ≼ ω)
5 rnct 9950 . . . . 5 ((𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ≼ ω → ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ≼ ω)
63, 4, 53syl 18 . . . 4 (𝜑 → ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ≼ ω)
7 vex 3500 . . . . . . . 8 𝑦 ∈ V
8 eqid 2824 . . . . . . . . 9 (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) = (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
98elrnmpt 5831 . . . . . . . 8 (𝑦 ∈ V → (𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ↔ ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}))
107, 9ax-mp 5 . . . . . . 7 (𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ↔ ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
1110biimpi 218 . . . . . 6 (𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
1211adantl 484 . . . . 5 ((𝜑𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
13 simp3 1134 . . . . . . . . 9 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
14 smfpimcc.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1514adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
16 smfpimcc.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1716ffvelrnda 6854 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
18 eqid 2824 . . . . . . . . . . . . 13 dom (𝐹𝑚) = dom (𝐹𝑚)
19 smfpimcc.j . . . . . . . . . . . . 13 𝐽 = (topGen‘ran (,))
20 smfpimcc.b . . . . . . . . . . . . 13 𝐵 = (SalGen‘𝐽)
21 smfpimcc.a . . . . . . . . . . . . . 14 (𝜑𝐴𝐵)
2221adantr 483 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐴𝐵)
23 eqid 2824 . . . . . . . . . . . . 13 ((𝐹𝑚) “ 𝐴) = ((𝐹𝑚) “ 𝐴)
2415, 17, 18, 19, 20, 22, 23smfpimbor1 43082 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → ((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)))
25 fvex 6686 . . . . . . . . . . . . . . . 16 (𝐹𝑚) ∈ V
2625dmex 7619 . . . . . . . . . . . . . . 15 dom (𝐹𝑚) ∈ V
2726a1i 11 . . . . . . . . . . . . . 14 (𝜑 → dom (𝐹𝑚) ∈ V)
28 elrest 16704 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ dom (𝐹𝑚) ∈ V) → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
2914, 27, 28syl2anc 586 . . . . . . . . . . . . 13 (𝜑 → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
3029adantr 483 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
3124, 30mpbid 234 . . . . . . . . . . 11 ((𝜑𝑚𝑍) → ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚)))
32 rabn0 4342 . . . . . . . . . . 11 ({𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅ ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚)))
3331, 32sylibr 236 . . . . . . . . . 10 ((𝜑𝑚𝑍) → {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
34333adant3 1128 . . . . . . . . 9 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
3513, 34eqnetrd 3086 . . . . . . . 8 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
36353exp 1115 . . . . . . 7 (𝜑 → (𝑚𝑍 → (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
3736rexlimdv 3286 . . . . . 6 (𝜑 → (∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
3837adantr 483 . . . . 5 ((𝜑𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → (∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
3912, 38mpd 15 . . . 4 ((𝜑𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → 𝑦 ≠ ∅)
406, 39axccd2 41502 . . 3 (𝜑 → ∃𝑓𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦)
41 nfv 1914 . . . . . . 7 𝑚𝜑
42 nfmpt1 5167 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
4342nfrn 5827 . . . . . . . 8 𝑚ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
44 nfv 1914 . . . . . . . 8 𝑚(𝑓𝑦) ∈ 𝑦
4543, 44nfralw 3228 . . . . . . 7 𝑚𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦
4641, 45nfan 1899 . . . . . 6 𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦)
471fvexi 6687 . . . . . 6 𝑍 ∈ V
4814adantr 483 . . . . . 6 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
49 fveq2 6673 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
50 id 22 . . . . . . . . 9 (𝑦 = 𝑤𝑦 = 𝑤)
5149, 50eleq12d 2910 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑤) ∈ 𝑤))
5251rspccva 3625 . . . . . . 7 ((∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦𝑤 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → (𝑓𝑤) ∈ 𝑤)
5352adantll 712 . . . . . 6 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) ∧ 𝑤 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → (𝑓𝑤) ∈ 𝑤)
54 eqid 2824 . . . . . 6 (𝑚𝑍 ↦ (𝑓‘{𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) = (𝑚𝑍 ↦ (𝑓‘{𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}))
5546, 47, 48, 53, 54smfpimcclem 43088 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))))
5655ex 415 . . . 4 (𝜑 → (∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)))))
5756exlimdv 1933 . . 3 (𝜑 → (∃𝑓𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)))))
5840, 57mpd 15 . 2 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))))
59 smfpimcc.1 . . . . . . . . 9 𝑛𝐹
60 nfcv 2980 . . . . . . . . 9 𝑛𝑚
6159, 60nffv 6683 . . . . . . . 8 𝑛(𝐹𝑚)
6261nfcnv 5752 . . . . . . 7 𝑛(𝐹𝑚)
63 nfcv 2980 . . . . . . 7 𝑛𝐴
6462, 63nfima 5940 . . . . . 6 𝑛((𝐹𝑚) “ 𝐴)
65 nfcv 2980 . . . . . . 7 𝑛(𝑚)
6661nfdm 5826 . . . . . . 7 𝑛dom (𝐹𝑚)
6765, 66nfin 4196 . . . . . 6 𝑛((𝑚) ∩ dom (𝐹𝑚))
6864, 67nfeq 2994 . . . . 5 𝑛((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))
69 nfv 1914 . . . . 5 𝑚((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))
70 fveq2 6673 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
7170cnveqd 5749 . . . . . . 7 (𝑚 = 𝑛(𝐹𝑚) = (𝐹𝑛))
7271imaeq1d 5931 . . . . . 6 (𝑚 = 𝑛 → ((𝐹𝑚) “ 𝐴) = ((𝐹𝑛) “ 𝐴))
73 fveq2 6673 . . . . . . 7 (𝑚 = 𝑛 → (𝑚) = (𝑛))
7470dmeqd 5777 . . . . . . 7 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
7573, 74ineq12d 4193 . . . . . 6 (𝑚 = 𝑛 → ((𝑚) ∩ dom (𝐹𝑚)) = ((𝑛) ∩ dom (𝐹𝑛)))
7672, 75eqeq12d 2840 . . . . 5 (𝑚 = 𝑛 → (((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)) ↔ ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
7768, 69, 76cbvralw 3444 . . . 4 (∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)) ↔ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)))
7877anbi2i 624 . . 3 ((:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))) ↔ (:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
7978exbii 1847 . 2 (∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))) ↔ ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
8058, 79sylib 220 1 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1083   = wceq 1536  wex 1779  wcel 2113  wnfc 2964  wne 3019  wral 3141  wrex 3142  {crab 3145  Vcvv 3497  cin 3938  c0 4294   class class class wbr 5069  cmpt 5149  ccnv 5557  dom cdm 5558  ran crn 5559  cima 5561  wf 6354  cfv 6358  (class class class)co 7159  ωcom 7583  cdom 8510  cuz 12246  (,)cioo 12741  t crest 16697  topGenctg 16714  SAlgcsalg 42600  SalGencsalgen 42604  SMblFncsmblfn 42984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-cc 9860  ax-ac2 9888  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-se 5518  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-oadd 8109  df-omul 8110  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-fin 8516  df-sup 8909  df-inf 8910  df-oi 8977  df-card 9371  df-acn 9374  df-ac 9545  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-ioo 12745  df-ico 12747  df-fl 13165  df-rest 16699  df-topgen 16720  df-top 21505  df-bases 21557  df-salg 42601  df-salgen 42605  df-smblfn 42985
This theorem is referenced by:  smfsuplem2  43093
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