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Theorem smflimsuplem2 46776
Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
smflimsuplem2.p 𝑚𝜑
smflimsuplem2.m (𝜑𝑀 ∈ ℤ)
smflimsuplem2.z 𝑍 = (ℤ𝑀)
smflimsuplem2.s (𝜑𝑆 ∈ SAlg)
smflimsuplem2.f (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem2.e 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem2.h 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem2.n (𝜑𝑛𝑍)
smflimsuplem2.r (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
smflimsuplem2.x (𝜑𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
Assertion
Ref Expression
smflimsuplem2 (𝜑𝑋 ∈ dom (𝐻𝑛))
Distinct variable groups:   𝑥,𝐹   𝑚,𝑀   𝑚,𝑋   𝑚,𝑍,𝑛,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝑆(𝑥,𝑚,𝑛)   𝐸(𝑥,𝑚,𝑛)   𝐹(𝑚,𝑛)   𝐻(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑛)   𝑋(𝑥,𝑛)

Proof of Theorem smflimsuplem2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 smflimsuplem2.x . . . 4 (𝜑𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
2 smflimsuplem2.p . . . . . 6 𝑚𝜑
3 eqid 2734 . . . . . 6 (ℤ𝑛) = (ℤ𝑛)
4 smflimsuplem2.n . . . . . . . . . . . . 13 (𝜑𝑛𝑍)
5 smflimsuplem2.z . . . . . . . . . . . . 13 𝑍 = (ℤ𝑀)
64, 5eleqtrdi 2848 . . . . . . . . . . . 12 (𝜑𝑛 ∈ (ℤ𝑀))
7 uzss 12898 . . . . . . . . . . . 12 (𝑛 ∈ (ℤ𝑀) → (ℤ𝑛) ⊆ (ℤ𝑀))
86, 7syl 17 . . . . . . . . . . 11 (𝜑 → (ℤ𝑛) ⊆ (ℤ𝑀))
98, 5sseqtrrdi 4046 . . . . . . . . . 10 (𝜑 → (ℤ𝑛) ⊆ 𝑍)
109adantr 480 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → (ℤ𝑛) ⊆ 𝑍)
11 simpr 484 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑛))
1210, 11sseldd 3995 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚𝑍)
13 smflimsuplem2.s . . . . . . . . . 10 (𝜑𝑆 ∈ SAlg)
1413adantr 480 . . . . . . . . 9 ((𝜑𝑚𝑍) → 𝑆 ∈ SAlg)
15 smflimsuplem2.f . . . . . . . . . 10 (𝜑𝐹:𝑍⟶(SMblFn‘𝑆))
1615ffvelcdmda 7103 . . . . . . . . 9 ((𝜑𝑚𝑍) → (𝐹𝑚) ∈ (SMblFn‘𝑆))
17 eqid 2734 . . . . . . . . 9 dom (𝐹𝑚) = dom (𝐹𝑚)
1814, 16, 17smff 46687 . . . . . . . 8 ((𝜑𝑚𝑍) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
1912, 18syldan 591 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑛)) → (𝐹𝑚):dom (𝐹𝑚)⟶ℝ)
20 iinss2 5061 . . . . . . . . 9 (𝑚 ∈ (ℤ𝑛) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ dom (𝐹𝑚))
2120adantl 481 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ⊆ dom (𝐹𝑚))
221adantr 480 . . . . . . . 8 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚))
2321, 22sseldd 3995 . . . . . . 7 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑋 ∈ dom (𝐹𝑚))
2419, 23ffvelcdmd 7104 . . . . . 6 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ ℝ)
25 nfmpt1 5255 . . . . . . . . 9 𝑚(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))
26 nfmpt1 5255 . . . . . . . . 9 𝑚(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))
27 eluzelz 12885 . . . . . . . . . 10 (𝑛 ∈ (ℤ𝑀) → 𝑛 ∈ ℤ)
286, 27syl 17 . . . . . . . . 9 (𝜑𝑛 ∈ ℤ)
29 eqid 2734 . . . . . . . . . . 11 (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))
302, 24, 29fmptdf 7136 . . . . . . . . . 10 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)):(ℤ𝑛)⟶ℝ)
3130ffnd 6737 . . . . . . . . 9 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) Fn (ℤ𝑛))
32 smflimsuplem2.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
33 nfcv 2902 . . . . . . . . . 10 𝑚(ℤ𝑀)
34 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑀)) → ((𝐹𝑚)‘𝑋) ∈ V)
3533, 2, 34mptfnd 45185 . . . . . . . . 9 (𝜑 → (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) Fn (ℤ𝑀))
3629a1i 11 . . . . . . . . . . 11 (𝜑 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
37 fvexd 6921 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝐹𝑚)‘𝑋) ∈ V)
3836, 37fvmpt2d 7028 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
3912, 5eleqtrdi 2848 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (ℤ𝑛)) → 𝑚 ∈ (ℤ𝑀))
40 eqid 2734 . . . . . . . . . . . 12 (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))
4140fvmpt2 7026 . . . . . . . . . . 11 ((𝑚 ∈ (ℤ𝑀) ∧ ((𝐹𝑚)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
4239, 37, 41syl2anc 584 . . . . . . . . . 10 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝐹𝑚)‘𝑋))
4338, 42eqtr4d 2777 . . . . . . . . 9 ((𝜑𝑚 ∈ (ℤ𝑛)) → ((𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))‘𝑚) = ((𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))‘𝑚))
442, 25, 26, 28, 31, 32, 35, 28, 43limsupequz 45678 . . . . . . . 8 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))))
455eqcomi 2743 . . . . . . . . . . 11 (ℤ𝑀) = 𝑍
4645mpteq1i 5243 . . . . . . . . . 10 (𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋)) = (𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))
4746fveq2i 6909 . . . . . . . . 9 (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋)))
4847a1i 11 . . . . . . . 8 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑀) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
4944, 48eqtrd 2774 . . . . . . 7 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) = (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))))
50 smflimsuplem2.r . . . . . . . 8 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ∈ ℝ)
5150renepnfd 11309 . . . . . . 7 (𝜑 → (lim sup‘(𝑚𝑍 ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
5249, 51eqnetrd 3005 . . . . . 6 (𝜑 → (lim sup‘(𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋))) ≠ +∞)
532, 3, 24, 52limsupubuzmpt 45674 . . . . 5 (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦)
54 uzid 12890 . . . . . . 7 (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ𝑛))
55 ne0i 4346 . . . . . . 7 (𝑛 ∈ (ℤ𝑛) → (ℤ𝑛) ≠ ∅)
5628, 54, 553syl 18 . . . . . 6 (𝜑 → (ℤ𝑛) ≠ ∅)
572, 56, 24supxrre3rnmpt 45378 . . . . 5 (𝜑 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ𝑛)((𝐹𝑚)‘𝑋) ≤ 𝑦))
5853, 57mpbird 257 . . . 4 (𝜑 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
591, 58jca 511 . . 3 (𝜑 → (𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
60 fveq2 6906 . . . . . . . . . 10 (𝑥 = 𝑦 → ((𝐹𝑚)‘𝑥) = ((𝐹𝑚)‘𝑦))
6160mpteq2dv 5249 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
6261rneqd 5951 . . . . . . . 8 (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)))
6362supeq1d 9483 . . . . . . 7 (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
6463eleq1d 2823 . . . . . 6 (𝑥 = 𝑦 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ))
6564cbvrabv 3443 . . . . 5 {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ}
6665eleq2i 2830 . . . 4 (𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↔ 𝑋 ∈ {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ})
67 fveq2 6906 . . . . . . . . 9 (𝑦 = 𝑋 → ((𝐹𝑚)‘𝑦) = ((𝐹𝑚)‘𝑋))
6867mpteq2dv 5249 . . . . . . . 8 (𝑦 = 𝑋 → (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
6968rneqd 5951 . . . . . . 7 (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)))
7069supeq1d 9483 . . . . . 6 (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ))
7170eleq1d 2823 . . . . 5 (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
7271elrab 3694 . . . 4 (𝑋 ∈ {𝑦 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ} ↔ (𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
7366, 72bitri 275 . . 3 (𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↔ (𝑋 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∧ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
7459, 73sylibr 234 . 2 (𝜑𝑋 ∈ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
75 id 22 . . . . 5 (𝜑𝜑)
76 smflimsuplem2.h . . . . . . 7 𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
7776a1i 11 . . . . . 6 (𝜑𝐻 = (𝑛𝑍 ↦ (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ))))
78 smflimsuplem2.e . . . . . . . . . 10 𝐸 = (𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
79 nfcv 2902 . . . . . . . . . . 11 𝑥𝑍
80 nfrab1 3453 . . . . . . . . . . 11 𝑥{𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
8179, 80nfmpt 5254 . . . . . . . . . 10 𝑥(𝑛𝑍 ↦ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
8278, 81nfcxfr 2900 . . . . . . . . 9 𝑥𝐸
83 nfcv 2902 . . . . . . . . 9 𝑥𝑛
8482, 83nffv 6916 . . . . . . . 8 𝑥(𝐸𝑛)
85 fvex 6919 . . . . . . . 8 (𝐸𝑛) ∈ V
8684, 85mptexf 45180 . . . . . . 7 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V
8786a1i 11 . . . . . 6 ((𝜑𝑛𝑍) → (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) ∈ V)
8877, 87fvmpt2d 7028 . . . . 5 ((𝜑𝑛𝑍) → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
8975, 4, 88syl2anc 584 . . . 4 (𝜑 → (𝐻𝑛) = (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
9089dmeqd 5918 . . 3 (𝜑 → dom (𝐻𝑛) = dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )))
91 nfcv 2902 . . . . 5 𝑦(𝐸𝑛)
92 nfcv 2902 . . . . 5 𝑦sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )
93 nfcv 2902 . . . . 5 𝑥sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < )
9484, 91, 92, 93, 63cbvmptf 5256 . . . 4 (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝑦 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ))
95 xrltso 13179 . . . . . 6 < Or ℝ*
9695supex 9500 . . . . 5 sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ V
9796a1i 11 . . . 4 ((𝜑𝑦 ∈ (𝐸𝑛)) → sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑦)), ℝ*, < ) ∈ V)
9894, 97dmmptd 6713 . . 3 (𝜑 → dom (𝑥 ∈ (𝐸𝑛) ↦ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < )) = (𝐸𝑛))
99 eqid 2734 . . . . 5 {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
100 fvex 6919 . . . . . . . . 9 (𝐹𝑚) ∈ V
101100dmex 7931 . . . . . . . 8 dom (𝐹𝑚) ∈ V
102101rgenw 3062 . . . . . . 7 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V
103102a1i 11 . . . . . 6 (𝜑 → ∀𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10456, 103iinexd 45072 . . . . 5 (𝜑 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∈ V)
10599, 104rabexd 5345 . . . 4 (𝜑 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
10678fvmpt2 7026 . . . 4 ((𝑛𝑍 ∧ {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V) → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
1074, 105, 106syl2anc 584 . . 3 (𝜑 → (𝐸𝑛) = {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
10890, 98, 1073eqtrrd 2779 . 2 (𝜑 → {𝑥 𝑚 ∈ (ℤ𝑛)dom (𝐹𝑚) ∣ sup(ran (𝑚 ∈ (ℤ𝑛) ↦ ((𝐹𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = dom (𝐻𝑛))
10974, 108eleqtrd 2840 1 (𝜑𝑋 ∈ dom (𝐻𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wnf 1779  wcel 2105  wne 2937  wral 3058  wrex 3067  {crab 3432  Vcvv 3477  wss 3962  c0 4338   ciin 4996   class class class wbr 5147  cmpt 5230  dom cdm 5688  ran crn 5689  wf 6558  cfv 6562  supcsup 9477  cr 11151  +∞cpnf 11289  *cxr 11291   < clt 11292  cle 11293  cz 12610  cuz 12875  lim supclsp 15502  SAlgcsalg 46263  SMblFncsmblfn 46650
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-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230
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-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  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-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  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-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  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-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-pm 8867  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-q 12988  df-ioo 13387  df-ico 13389  df-fz 13544  df-fl 13828  df-ceil 13829  df-limsup 15503  df-smblfn 46651
This theorem is referenced by:  smflimsuplem7  46781
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