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Theorem issmfgt 39643
 Description: The predicate "𝐹 is b 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 39619 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 39618 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1830 . . . . . . 7 𝑏𝜑
8 nfv 1830 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1816 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
102, 5restuni4 38336 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) = 𝐷)
1110eqcomd 2616 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = (𝑆t 𝐷))
1211rabeqd 38304 . . . . . . . . 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 1830 . . . . . . . . . . 11 𝑦𝜑
15 nfv 1830 . . . . . . . . . . 11 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1614, 15nfan 1816 . . . . . . . . . 10 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
17 nfv 1830 . . . . . . . . . 10 𝑦 𝑏 ∈ ℝ
1816, 17nfan 1816 . . . . . . . . 9 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19 nfv 1830 . . . . . . . . 9 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
201uniexd 38310 . . . . . . . . . . . . . 14 (𝜑 𝑆 ∈ V)
2120adantr 480 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
22 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
2321, 22ssexd 4733 . . . . . . . . . . . 12 ((𝜑𝐷 𝑆) → 𝐷 ∈ V)
245, 23syldan 486 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 ∈ V)
25 eqid 2610 . . . . . . . . . . 11 (𝑆t 𝐷) = (𝑆t 𝐷)
262, 24, 25subsalsal 39253 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2726adantr 480 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
28 eqid 2610 . . . . . . . . 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 2690 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦𝐷)
3329, 32ffvelrnd 6268 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ)
3433rexrd 9968 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
3534adantlr 747 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
362, 4issmfle 39632 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))))
373, 36mpbid 221 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
3837simp3d 1068 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
3910rabeqd 38304 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐})
4039eleq1d 2672 . . . . . . . . . . . . . 14 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ({𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4140ralbidv 2969 . . . . . . . . . . . . 13 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ↔ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷)))
4238, 41mpbird 246 . . . . . . . . . . . 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 2914 . . . . . . . . . . 11 ((∀𝑐 ∈ ℝ {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4643, 44, 45syl2anc 691 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
4746adantlr 747 . . . . . . . . 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 39597 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5013, 49eqeltrd 2688 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5150ex 449 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
529, 51ralrimi 2940 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
535, 6, 523jca 1235 . . . 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 1830 . . . . . . 7 𝑦 𝐷 𝑆
56 nfv 1830 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
57 nfcv 2751 . . . . . . . 8 𝑦
58 nfrab1 3099 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)}
59 nfcv 2751 . . . . . . . . 9 𝑦(𝑆t 𝐷)
6058, 59nfel 2763 . . . . . . . 8 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6157, 60nfral 2929 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6255, 56, 61nf3an 1819 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
6314, 62nfan 1816 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
64 nfv 1830 . . . . . . 7 𝑏 𝐷 𝑆
65 nfv 1830 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
66 nfra1 2925 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6764, 65, 66nf3an 1819 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
687, 67nfan 1816 . . . . 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 1060 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐷 𝑆)
71 simpr2 1061 . . . . 5 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹:𝐷⟶ℝ)
72 simpr3 1062 . . . . 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 39642 . . . 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 201 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
76 breq1 4586 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑦)))
7776rabbidv 3164 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦𝐷𝑎 < (𝐹𝑦)})
78 fveq2 6103 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7978breq2d 4595 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑥)))
8079cbvrabv 3172 . . . . . . . 8 {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)}
8180a1i 11 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑎 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8277, 81eqtrd 2644 . . . . . 6 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑥𝐷𝑎 < (𝐹𝑥)})
8382eleq1d 2672 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷)))
8483cbvralv 3147 . . . 4 (∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷))
85843anbi3i 1248 . . 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 267 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   ↾t crest 15904  SAlgcsalg 39204  SMblFncsmblfn 39586 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cc 9140  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-acn 8651  df-ac 8822  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-ioo 12050  df-ico 12052  df-fl 12455  df-rest 15906  df-salg 39205  df-smblfn 39587 This theorem is referenced by:  issmfgtd  39647  smfpreimagt  39648
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