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Theorem smfpimcc 41335
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 39546 . . . . . 6 𝑍 ≼ ω
32a1i 11 . . . . 5 (𝜑𝑍 ≼ ω)
4 mptct 9398 . . . . 5 (𝑍 ≼ ω → (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) ≼ ω)
5 rnct 9385 . . . . 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 3234 . . . . . . . 8 𝑦 ∈ V
8 eqid 2651 . . . . . . . . 9 (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) = (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
98elrnmpt 5404 . . . . . . . 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 206 . . . . . 6 (𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
1211adantl 481 . . . . 5 ((𝜑𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → ∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
13 simp3 1083 . . . . . . . . 9 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
14 smfpimcc.s . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ SAlg)
1514adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
16 smfpimcc.f . . . . . . . . . . . . . 14 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1716ffvelrnda 6399 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
18 eqid 2651 . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 13 ((𝜑𝑚𝑍) → 𝐴𝐵)
23 eqid 2651 . . . . . . . . . . . . 13 ((𝐹𝑚) “ 𝐴) = ((𝐹𝑚) “ 𝐴)
2415, 17, 18, 19, 20, 22, 23smfpimbor1 41328 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → ((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)))
25 fvex 6239 . . . . . . . . . . . . . . . 16 (𝐹𝑚) ∈ V
2625dmex 7141 . . . . . . . . . . . . . . 15 dom (𝐹𝑚) ∈ V
2726a1i 11 . . . . . . . . . . . . . 14 (𝜑 → dom (𝐹𝑚) ∈ V)
28 elrest 16135 . . . . . . . . . . . . . 14 ((𝑆 ∈ SAlg ∧ dom (𝐹𝑚) ∈ V) → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
2914, 27, 28syl2anc 694 . . . . . . . . . . . . 13 (𝜑 → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
3029adantr 480 . . . . . . . . . . . 12 ((𝜑𝑚𝑍) → (((𝐹𝑚) “ 𝐴) ∈ (𝑆t dom (𝐹𝑚)) ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))))
3124, 30mpbid 222 . . . . . . . . . . 11 ((𝜑𝑚𝑍) → ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚)))
32 rabn0 3991 . . . . . . . . . . 11 ({𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅ ↔ ∃𝑠𝑆 ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚)))
3331, 32sylibr 224 . . . . . . . . . 10 ((𝜑𝑚𝑍) → {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
34333adant3 1101 . . . . . . . . 9 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} ≠ ∅)
3513, 34eqnetrd 2890 . . . . . . . 8 ((𝜑𝑚𝑍𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}) → 𝑦 ≠ ∅)
36353exp 1283 . . . . . . 7 (𝜑 → (𝑚𝑍 → (𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅)))
3736rexlimdv 3059 . . . . . 6 (𝜑 → (∃𝑚𝑍 𝑦 = {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))} → 𝑦 ≠ ∅))
3837adantr 480 . . . . 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 39744 . . 3 (𝜑 → ∃𝑓𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦)
41 nfv 1883 . . . . . . 7 𝑚𝜑
42 nfmpt1 4780 . . . . . . . . 9 𝑚(𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
4342nfrn 5400 . . . . . . . 8 𝑚ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})
44 nfv 1883 . . . . . . . 8 𝑚(𝑓𝑦) ∈ 𝑦
4543, 44nfral 2974 . . . . . . 7 𝑚𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦
4641, 45nfan 1868 . . . . . 6 𝑚(𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦)
47 fvex 6239 . . . . . . 7 (ℤ𝑀) ∈ V
481, 47eqeltri 2726 . . . . . 6 𝑍 ∈ V
4914adantr 480 . . . . . 6 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) → 𝑆 ∈ SAlg)
50 fveq2 6229 . . . . . . . . 9 (𝑦 = 𝑤 → (𝑓𝑦) = (𝑓𝑤))
51 id 22 . . . . . . . . 9 (𝑦 = 𝑤𝑦 = 𝑤)
5250, 51eleq12d 2724 . . . . . . . 8 (𝑦 = 𝑤 → ((𝑓𝑦) ∈ 𝑦 ↔ (𝑓𝑤) ∈ 𝑤))
5352rspccva 3339 . . . . . . 7 ((∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦𝑤 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → (𝑓𝑤) ∈ 𝑤)
5453adantll 750 . . . . . 6 (((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) ∧ 𝑤 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) → (𝑓𝑤) ∈ 𝑤)
55 eqid 2651 . . . . . 6 (𝑚𝑍 ↦ (𝑓‘{𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})) = (𝑚𝑍 ↦ (𝑓‘{𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))}))
5646, 48, 49, 54, 55smfpimcclem 41334 . . . . 5 ((𝜑 ∧ ∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦) → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))))
5756ex 449 . . . 4 (𝜑 → (∀𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)))))
5857exlimdv 1901 . . 3 (𝜑 → (∃𝑓𝑦 ∈ ran (𝑚𝑍 ↦ {𝑠𝑆 ∣ ((𝐹𝑚) “ 𝐴) = (𝑠 ∩ dom (𝐹𝑚))})(𝑓𝑦) ∈ 𝑦 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)))))
5940, 58mpd 15 . 2 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))))
60 smfpimcc.1 . . . . . . . . 9 𝑛𝐹
61 nfcv 2793 . . . . . . . . 9 𝑛𝑚
6260, 61nffv 6236 . . . . . . . 8 𝑛(𝐹𝑚)
6362nfcnv 5333 . . . . . . 7 𝑛(𝐹𝑚)
64 nfcv 2793 . . . . . . 7 𝑛𝐴
6563, 64nfima 5509 . . . . . 6 𝑛((𝐹𝑚) “ 𝐴)
66 nfcv 2793 . . . . . . 7 𝑛(𝑚)
6762nfdm 5399 . . . . . . 7 𝑛dom (𝐹𝑚)
6866, 67nfin 3853 . . . . . 6 𝑛((𝑚) ∩ dom (𝐹𝑚))
6965, 68nfeq 2805 . . . . 5 𝑛((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))
70 nfv 1883 . . . . 5 𝑚((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))
71 fveq2 6229 . . . . . . . 8 (𝑚 = 𝑛 → (𝐹𝑚) = (𝐹𝑛))
7271cnveqd 5330 . . . . . . 7 (𝑚 = 𝑛(𝐹𝑚) = (𝐹𝑛))
7372imaeq1d 5500 . . . . . 6 (𝑚 = 𝑛 → ((𝐹𝑚) “ 𝐴) = ((𝐹𝑛) “ 𝐴))
74 fveq2 6229 . . . . . . 7 (𝑚 = 𝑛 → (𝑚) = (𝑛))
7571dmeqd 5358 . . . . . . 7 (𝑚 = 𝑛 → dom (𝐹𝑚) = dom (𝐹𝑛))
7674, 75ineq12d 3848 . . . . . 6 (𝑚 = 𝑛 → ((𝑚) ∩ dom (𝐹𝑚)) = ((𝑛) ∩ dom (𝐹𝑛)))
7773, 76eqeq12d 2666 . . . . 5 (𝑚 = 𝑛 → (((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)) ↔ ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
7869, 70, 77cbvral 3197 . . . 4 (∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚)) ↔ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛)))
7978anbi2i 730 . . 3 ((:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))) ↔ (:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
8079exbii 1814 . 2 (∃(:𝑍𝑆 ∧ ∀𝑚𝑍 ((𝐹𝑚) “ 𝐴) = ((𝑚) ∩ dom (𝐹𝑚))) ↔ ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
8159, 80sylib 208 1 (𝜑 → ∃(:𝑍𝑆 ∧ ∀𝑛𝑍 ((𝐹𝑛) “ 𝐴) = ((𝑛) ∩ dom (𝐹𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wnfc 2780  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  cin 3606  c0 3948   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  ωcom 7107  cdom 7995  cuz 11725  (,)cioo 12213  t crest 16128  topGenctg 16145  SAlgcsalg 40846  SalGencsalgen 40850  SMblFncsmblfn 41230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cc 9295  ax-ac2 9323  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-acn 8806  df-ac 8977  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-ioo 12217  df-ico 12219  df-fl 12633  df-rest 16130  df-topgen 16151  df-top 20747  df-bases 20798  df-salg 40847  df-salgen 40851  df-smblfn 41231
This theorem is referenced by:  smfsuplem2  41339
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