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Theorem issmfgt 44987
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 481 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝑆 ∈ SAlg)
3 simpr 485 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfgt.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 44964 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 44963 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1917 . . . . . . 7 𝑏𝜑
8 nfv 1917 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1902 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
102, 5restuni4 43321 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) = 𝐷)
1110eqcomd 2742 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = (𝑆t 𝐷))
1211rabeqdv 3422 . . . . . . . . 9 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)})
1312adantr 481 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)})
14 nfv 1917 . . . . . . . . . . 11 𝑦𝜑
15 nfv 1917 . . . . . . . . . . 11 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1614, 15nfan 1902 . . . . . . . . . 10 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
17 nfv 1917 . . . . . . . . . 10 𝑦 𝑏 ∈ ℝ
1816, 17nfan 1902 . . . . . . . . 9 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19 nfv 1917 . . . . . . . . 9 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
201uniexd 7679 . . . . . . . . . . . . . 14 (𝜑 𝑆 ∈ V)
2120adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
22 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
2321, 22ssexd 5281 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
245, 23syldan 591 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25 eqid 2736 . . . . . . . . . . 11 (𝑆t 𝐷) = (𝑆t 𝐷)
262, 24, 25subsalsal 44590 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2726adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
28 eqid 2736 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
296adantr 481 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝐹:𝐷⟶ℝ)
30 simpr 485 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦 (𝑆t 𝐷))
3110adantr 481 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝑆t 𝐷) = 𝐷)
3230, 31eleqtrd 2840 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦𝐷)
3329, 32ffvelcdmd 7036 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ)
3433rexrd 11205 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
3534adantlr 713 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
362, 4issmfle 44976 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))))
373, 36mpbid 231 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
3837simp3d 1144 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
3910rabeqdv 3422 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐})
4039eleq1d 2822 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ({𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4140ralbidv 3174 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4238, 41mpbird 256 . . . . . . . . . . . 12 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4342adantr 481 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → ∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
44 simpr 485 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
45 rspa 3231 . . . . . . . . . . 11 ((∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4643, 44, 45syl2anc 584 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4746adantlr 713 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
48 simpr 485 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
4918, 19, 27, 28, 35, 47, 48salpreimalegt 44940 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5013, 49eqeltrd 2838 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5150ex 413 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
529, 51ralrimi 3240 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
535, 6, 523jca 1128 . . . 4 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
5453ex 413 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
55 nfv 1917 . . . . . . 7 𝑦 𝐷 𝑆
56 nfv 1917 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
57 nfcv 2907 . . . . . . . 8 𝑦
58 nfrab1 3426 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)}
59 nfcv 2907 . . . . . . . . 9 𝑦(𝑆t 𝐷)
6058, 59nfel 2921 . . . . . . . 8 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6157, 60nfralw 3294 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6255, 56, 61nf3an 1904 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
6314, 62nfan 1902 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
64 nfv 1917 . . . . . . 7 𝑏 𝐷 𝑆
65 nfv 1917 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
66 nfra1 3267 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6764, 65, 66nf3an 1904 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
687, 67nfan 1902 . . . . 5 𝑏(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
691adantr 481 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝑆 ∈ SAlg)
70 simpr1 1194 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
71 simpr2 1195 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
72 simpr3 1196 . . . . 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 44986 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
7473ex 413 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
7554, 74impbid 211 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
76 breq1 5108 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑦)))
7776rabbidv 3415 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦𝐷𝑎 < (𝐹𝑦)})
78 fveq2 6842 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7978breq2d 5117 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑥)))
8079cbvrabv 3417 . . . . . . . 8 {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)}
8180a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8277, 81eqtrd 2776 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8382eleq1d 2822 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷)))
8483cbvralvw 3225 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))
85843anbi3i 1159 . . 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 278 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  wss 3910   cuni 4865   class class class wbr 5105  dom cdm 5633  wf 6492  cfv 6496  (class class class)co 7357  cr 11050  *cxr 11188   < clt 11189  cle 11190  t crest 17302  SAlgcsalg 44539  SMblFncsmblfn 44926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-ioo 13268  df-ico 13270  df-fl 13697  df-rest 17304  df-salg 44540  df-smblfn 44927
This theorem is referenced by:  issmfgtd  44992  smfpreimagt  44993
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