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Theorem issmfgt 45458
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 45435 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 45434 . . . . 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 43795 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) = 𝐷)
1110eqcomd 2738 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = (𝑆t 𝐷))
1211rabeqdv 3447 . . . . . . . . 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 7728 . . . . . . . . . . . . . 14 (𝜑 𝑆 ∈ V)
2120adantr 481 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
22 simpr 485 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
2321, 22ssexd 5323 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
245, 23syldan 591 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25 eqid 2732 . . . . . . . . . . 11 (𝑆t 𝐷) = (𝑆t 𝐷)
262, 24, 25subsalsal 45061 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2726adantr 481 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
28 eqid 2732 . . . . . . . . 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 2835 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦𝐷)
3329, 32ffvelcdmd 7084 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ)
3433rexrd 11260 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
3534adantlr 713 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
362, 4issmfle 45447 . . . . . . . . . . . . . . 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 3447 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐})
4039eleq1d 2818 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ({𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4140ralbidv 3177 . . . . . . . . . . . . 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 3245 . . . . . . . . . . 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 45411 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5013, 49eqeltrd 2833 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5150ex 413 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
529, 51ralrimi 3254 . . . . 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 2903 . . . . . . . 8 𝑦
58 nfrab1 3451 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)}
59 nfcv 2903 . . . . . . . . 9 𝑦(𝑆t 𝐷)
6058, 59nfel 2917 . . . . . . . 8 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6157, 60nfralw 3308 . . . . . . 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 3281 . . . . . . 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 45457 . . . 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 5150 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑦)))
7776rabbidv 3440 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦𝐷𝑎 < (𝐹𝑦)})
78 fveq2 6888 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7978breq2d 5159 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑥)))
8079cbvrabv 3442 . . . . . . . 8 {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)}
8180a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8277, 81eqtrd 2772 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8382eleq1d 2818 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷)))
8483cbvralvw 3234 . . . 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 3061  {crab 3432  Vcvv 3474  wss 3947   cuni 4907   class class class wbr 5147  dom cdm 5675  wf 6536  cfv 6540  (class class class)co 7405  cr 11105  *cxr 11243   < clt 11244  cle 11245  t crest 17362  SAlgcsalg 45010  SMblFncsmblfn 45397
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cc 10426  ax-ac2 10454  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-card 9930  df-acn 9933  df-ac 10107  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-ioo 13324  df-ico 13326  df-fl 13753  df-rest 17364  df-salg 45011  df-smblfn 45398
This theorem is referenced by:  issmfgtd  45463  smfpreimagt  45464
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