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Theorem smfpimcc 41808
 Description: Given a countable set of sigma-measurable functions, and a Borel set 𝐴 there exists a choice function ℎ 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 40049 . . . . . 6 𝑍 ≼ ω
32a1i 11 . . . . 5 (𝜑𝑍 ≼ ω)
4 mptct 9675 . . . . 5 (𝑍 ≼ ω → (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ≼ ω)
5 rnct 9662 . . . . 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 3417 . . . . . . . 8 𝑦 ∈ V
8 eqid 2825 . . . . . . . . 9 (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) = (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
98elrnmpt 5605 . . . . . . . 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 208 . . . . . 6 (𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
1211adantl 475 . . . . 5 ((𝜑𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
13 simp3 1174 . . . . . . . . 9 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
14 smfpimcc.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1514adantr 474 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
16 smfpimcc.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1716ffvelrnda 6608 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
18 eqid 2825 . . . . . . . . . . . . 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 474 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐴𝐵)
23 eqid 2825 . . . . . . . . . . . . 13 ((𝐹𝑚) “ 𝐴) = ((𝐹𝑚) “ 𝐴)
2415, 17, 18, 19, 20, 22, 23smfpimbor1 41801 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → ((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)))
25 fvex 6446 . . . . . . . . . . . . . . . 16 (𝐹𝑚) ∈ V
2625dmex 7361 . . . . . . . . . . . . . . 15 dom (𝐹𝑚) ∈ V
2726a1i 11 . . . . . . . . . . . . . 14 (𝜑 → dom (𝐹𝑚) ∈ V)
28 elrest 16441 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ dom (𝐹𝑚) ∈ V) → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
2914, 27, 28syl2anc 581 . . . . . . . . . . . . 13 (𝜑 → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
3029adantr 474 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
3124, 30mpbid 224 . . . . . . . . . . 11 ((𝜑𝑚𝑍) → ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚)))
32 rabn0 4187 . . . . . . . . . . 11 ({𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅ ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚)))
3331, 32sylibr 226 . . . . . . . . . 10 ((𝜑𝑚𝑍) → {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
34333adant3 1168 . . . . . . . . 9 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
3513, 34eqnetrd 3066 . . . . . . . 8 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
36353exp 1154 . . . . . . 7 (𝜑 → (𝑚𝑍 → (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
3736rexlimdv 3239 . . . . . 6 (𝜑 → (∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
3837adantr 474 . . . . 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 40232 . . 3 (𝜑 → ∃𝑓𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦)
41 nfv 2015 . . . . . . 7 𝑚𝜑
42 nfmpt1 4970 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
4342nfrn 5601 . . . . . . . 8 𝑚ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
44 nfv 2015 . . . . . . . 8 𝑚(𝑓𝑦) ∈ 𝑦
4543, 44nfral 3154 . . . . . . 7 𝑚𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦
4641, 45nfan 2004 . . . . . 6 𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦)
471fvexi 6447 . . . . . 6 𝑍 ∈ V
4814adantr 474 . . . . . 6 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
49 fveq2 6433 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
50 id 22 . . . . . . . . 9 (𝑦 = 𝑤𝑦 = 𝑤)
5149, 50eleq12d 2900 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑤) ∈ 𝑤))
5251rspccva 3525 . . . . . . 7 ((∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦𝑤 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → (𝑓𝑤) ∈ 𝑤)
5352adantll 707 . . . . . 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 2825 . . . . . 6 (𝑚𝑍 ↦ (𝑓‘{𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) = (𝑚𝑍 ↦ (𝑓‘{𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}))
5546, 47, 48, 53, 54smfpimcclem 41807 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))))
5655ex 403 . . . 4 (𝜑 → (∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)))))
5756exlimdv 2034 . . 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 2969 . . . . . . . . 9 𝑛𝑚
6159, 60nffv 6443 . . . . . . . 8 𝑛(𝐹𝑚)
6261nfcnv 5533 . . . . . . 7 𝑛(𝐹𝑚)
63 nfcv 2969 . . . . . . 7 𝑛𝐴
6462, 63nfima 5715 . . . . . 6 𝑛((𝐹𝑚) “ 𝐴)
65 nfcv 2969 . . . . . . 7 𝑛(𝑚)
6661nfdm 5600 . . . . . . 7 𝑛dom (𝐹𝑚)
6765, 66nfin 4045 . . . . . 6 𝑛((𝑚) ∩ dom (𝐹𝑚))
6864, 67nfeq 2981 . . . . 5 𝑛((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))
69 nfv 2015 . . . . 5 𝑚((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))
70 fveq2 6433 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
7170cnveqd 5530 . . . . . . 7 (𝑚 = 𝑛(𝐹𝑚) = (𝐹𝑛))
7271imaeq1d 5706 . . . . . 6 (𝑚 = 𝑛 → ((𝐹𝑚) “ 𝐴) = ((𝐹𝑛) “ 𝐴))
73 fveq2 6433 . . . . . . 7 (𝑚 = 𝑛 → (𝑚) = (𝑛))
7470dmeqd 5558 . . . . . . 7 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
7573, 74ineq12d 4042 . . . . . 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, 76cbvral 3379 . . . 4 (∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)) ↔ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)))
7877anbi2i 618 . . 3 ((:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))) ↔ (:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
7978exbii 1949 . 2 (∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))) ↔ ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
8058, 79sylib 210 1 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658  ∃wex 1880   ∈ wcel 2166  Ⅎwnfc 2956   ≠ wne 2999  ∀wral 3117  ∃wrex 3118  {crab 3121  Vcvv 3414   ∩ cin 3797  ∅c0 4144   class class class wbr 4873   ↦ cmpt 4952  ◡ccnv 5341  dom cdm 5342  ran crn 5343   “ cima 5345  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  ωcom 7326   ≼ cdom 8220  ℤ≥cuz 11968  (,)cioo 12463   ↾t crest 16434  topGenctg 16451  SAlgcsalg 41319  SalGencsalgen 41323  SMblFncsmblfn 41703 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cc 9572  ax-ac2 9600  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329  ax-pre-sup 10330 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-omul 7831  df-er 8009  df-map 8124  df-pm 8125  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-inf 8618  df-oi 8684  df-card 9078  df-acn 9081  df-ac 9252  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-q 12072  df-rp 12113  df-ioo 12467  df-ico 12469  df-fl 12888  df-rest 16436  df-topgen 16457  df-top 21069  df-bases 21121  df-salg 41320  df-salgen 41324  df-smblfn 41704 This theorem is referenced by:  smfsuplem2  41812
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