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Theorem issmfgt 41286
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all left-open intervals unbounded above are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be b subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (iii) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
issmfgt.s (𝜑𝑆 ∈ SAlg)
issmfgt.d 𝐷 = dom 𝐹
Assertion
Ref Expression
issmfgt (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))))
Distinct variable groups:   𝐷,𝑎,𝑥   𝐹,𝑎,𝑥   𝑆,𝑎
Allowed substitution hints:   𝜑(𝑥,𝑎)   𝑆(𝑥)

Proof of Theorem issmfgt
Dummy variables 𝑏 𝑦 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issmfgt.s . . . . . . 7 (𝜑𝑆 ∈ SAlg)
21adantr 480 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 476 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfgt.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 41263 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 41262 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1883 . . . . . . 7 𝑏𝜑
8 nfv 1883 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1868 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
102, 5restuni4 39618 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) = 𝐷)
1110eqcomd 2657 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = (𝑆t 𝐷))
1211rabeqd 39590 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)})
1312adantr 480 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)})
14 nfv 1883 . . . . . . . . . . 11 𝑦𝜑
15 nfv 1883 . . . . . . . . . . 11 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1614, 15nfan 1868 . . . . . . . . . 10 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
17 nfv 1883 . . . . . . . . . 10 𝑦 𝑏 ∈ ℝ
1816, 17nfan 1868 . . . . . . . . 9 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19 nfv 1883 . . . . . . . . 9 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
201uniexd 39595 . . . . . . . . . . . . . 14 (𝜑 𝑆 ∈ V)
2120adantr 480 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
22 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
2321, 22ssexd 4838 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
245, 23syldan 486 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25 eqid 2651 . . . . . . . . . . 11 (𝑆t 𝐷) = (𝑆t 𝐷)
262, 24, 25subsalsal 40895 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2726adantr 480 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
28 eqid 2651 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
296adantr 480 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝐹:𝐷⟶ℝ)
30 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦 (𝑆t 𝐷))
3110adantr 480 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝑆t 𝐷) = 𝐷)
3230, 31eleqtrd 2732 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦𝐷)
3329, 32ffvelrnd 6400 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ)
3433rexrd 10127 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
3534adantlr 751 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
362, 4issmfle 41275 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))))
373, 36mpbid 222 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
3837simp3d 1095 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
3910rabeqd 39590 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐})
4039eleq1d 2715 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ({𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4140ralbidv 3015 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4238, 41mpbird 247 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4342adantr 480 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → ∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
44 simpr 476 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
45 rspa 2959 . . . . . . . . . . 11 ((∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4643, 44, 45syl2anc 694 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4746adantlr 751 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
48 simpr 476 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
4918, 19, 27, 28, 35, 47, 48salpreimalegt 41241 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5013, 49eqeltrd 2730 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5150ex 449 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
529, 51ralrimi 2986 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
535, 6, 523jca 1261 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
5453ex 449 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
55 nfv 1883 . . . . . . 7 𝑦 𝐷 𝑆
56 nfv 1883 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
57 nfcv 2793 . . . . . . . 8 𝑦
58 nfrab1 3152 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)}
59 nfcv 2793 . . . . . . . . 9 𝑦(𝑆t 𝐷)
6058, 59nfel 2806 . . . . . . . 8 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6157, 60nfral 2974 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6255, 56, 61nf3an 1871 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
6314, 62nfan 1868 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
64 nfv 1883 . . . . . . 7 𝑏 𝐷 𝑆
65 nfv 1883 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
66 nfra1 2970 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6764, 65, 66nf3an 1871 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
687, 67nfan 1868 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
691adantr 480 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
70 simpr1 1087 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
71 simpr2 1088 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
72 simpr3 1089 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
7363, 68, 69, 4, 70, 71, 72issmfgtlem 41285 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
7473ex 449 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
7554, 74impbid 202 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
76 breq1 4688 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑦)))
7776rabbidv 3220 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦𝐷𝑎 < (𝐹𝑦)})
78 fveq2 6229 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7978breq2d 4697 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑥)))
8079cbvrabv 3230 . . . . . . . 8 {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)}
8180a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8277, 81eqtrd 2685 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8382eleq1d 2715 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷)))
8483cbvralv 3201 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))
85843anbi3i 1274 . . 3 ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷)))
8685a1i 11 . 2 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))))
8775, 86bitrd 268 1 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  wss 3607   cuni 4468   class class class wbr 4685  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  cr 9973  *cxr 10111   < clt 10112  cle 10113  t crest 16128  SAlgcsalg 40846  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-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-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-salg 40847  df-smblfn 41231
This theorem is referenced by:  issmfgtd  41290  smfpreimagt  41291
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