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Theorem issmfgt 46776
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 484 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹 ∈ (SMblFn‘𝑆))
4 issmfgt.d . . . . . 6 𝐷 = dom 𝐹
52, 3, 4smfdmss 46753 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 𝑆)
62, 3, 4smff 46752 . . . . 5 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐹:𝐷⟶ℝ)
7 nfv 1913 . . . . . . 7 𝑏𝜑
8 nfv 1913 . . . . . . 7 𝑏 𝐹 ∈ (SMblFn‘𝑆)
97, 8nfan 1898 . . . . . 6 𝑏(𝜑𝐹 ∈ (SMblFn‘𝑆))
102, 5restuni4 45131 . . . . . . . . . . 11 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) = 𝐷)
1110eqcomd 2742 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → 𝐷 = (𝑆t 𝐷))
1211rabeqdv 3451 . . . . . . . . 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 1913 . . . . . . . . . . 11 𝑦𝜑
15 nfv 1913 . . . . . . . . . . 11 𝑦 𝐹 ∈ (SMblFn‘𝑆)
1614, 15nfan 1898 . . . . . . . . . 10 𝑦(𝜑𝐹 ∈ (SMblFn‘𝑆))
17 nfv 1913 . . . . . . . . . 10 𝑦 𝑏 ∈ ℝ
1816, 17nfan 1898 . . . . . . . . 9 𝑦((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
19 nfv 1913 . . . . . . . . 9 𝑐((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ)
201uniexd 7763 . . . . . . . . . . . . . 14 (𝜑 𝑆 ∈ V)
2120adantr 480 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝑆 ∈ V)
22 simpr 484 . . . . . . . . . . . . 13 ((𝜑𝐷 𝑆) → 𝐷 𝑆)
2321, 22ssexd 5323 . . . . . . . . . . . 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 46379 . . . . . . . . . 10 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑆t 𝐷) ∈ SAlg)
2726adantr 480 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → (𝑆t 𝐷) ∈ SAlg)
28 eqid 2736 . . . . . . . . 9 (𝑆t 𝐷) = (𝑆t 𝐷)
296adantr 480 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝐹:𝐷⟶ℝ)
30 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦 (𝑆t 𝐷))
3110adantr 480 . . . . . . . . . . . . 13 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝑆t 𝐷) = 𝐷)
3230, 31eleqtrd 2842 . . . . . . . . . . . 12 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → 𝑦𝐷)
3329, 32ffvelcdmd 7104 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ)
3433rexrd 11312 . . . . . . . . . 10 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
3534adantlr 715 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑦 (𝑆t 𝐷)) → (𝐹𝑦) ∈ ℝ*)
362, 4issmfle 46765 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑐 ∈ ℝ {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))))
373, 36mpbid 232 . . . . . . . . . . . . . 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 3451 . . . . . . . . . . . . . . 15 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} = {𝑦𝐷 ∣ (𝐹𝑦) ≤ 𝑐})
4039eleq1d 2825 . . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 484 . . . . . . . . . . 11 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
45 rspa 3247 . . . . . . . . . . 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 715 . . . . . . . . 9 ((((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) ∧ 𝑐 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ (𝐹𝑦) ≤ 𝑐} ∈ (𝑆t 𝐷))
48 simpr 484 . . . . . . . . 9 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → 𝑏 ∈ ℝ)
4918, 19, 27, 28, 35, 47, 48salpreimalegt 46729 . . . . . . . 8 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦 (𝑆t 𝐷) ∣ 𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5013, 49eqeltrd 2840 . . . . . . 7 (((𝜑𝐹 ∈ (SMblFn‘𝑆)) ∧ 𝑏 ∈ ℝ) → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
5150ex 412 . . . . . 6 ((𝜑𝐹 ∈ (SMblFn‘𝑆)) → (𝑏 ∈ ℝ → {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
529, 51ralrimi 3256 . . . . 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 412 . . 3 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) → (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
55 nfv 1913 . . . . . . 7 𝑦 𝐷 𝑆
56 nfv 1913 . . . . . . 7 𝑦 𝐹:𝐷⟶ℝ
57 nfcv 2904 . . . . . . . 8 𝑦
58 nfrab1 3456 . . . . . . . . 9 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)}
59 nfcv 2904 . . . . . . . . 9 𝑦(𝑆t 𝐷)
6058, 59nfel 2919 . . . . . . . 8 𝑦{𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6157, 60nfralw 3310 . . . . . . 7 𝑦𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6255, 56, 61nf3an 1900 . . . . . 6 𝑦(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
6314, 62nfan 1898 . . . . 5 𝑦(𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)))
64 nfv 1913 . . . . . . 7 𝑏 𝐷 𝑆
65 nfv 1913 . . . . . . 7 𝑏 𝐹:𝐷⟶ℝ
66 nfra1 3283 . . . . . . 7 𝑏𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)
6764, 65, 66nf3an 1900 . . . . . 6 𝑏(𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))
687, 67nfan 1898 . . . . 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 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 46775 . . . 4 ((𝜑 ∧ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))) → 𝐹 ∈ (SMblFn‘𝑆))
7473ex 412 . . 3 (𝜑 → ((𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷)) → 𝐹 ∈ (SMblFn‘𝑆)))
7554, 74impbid 212 . 2 (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 𝑆𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷))))
76 breq1 5145 . . . . . . . 8 (𝑏 = 𝑎 → (𝑏 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑦)))
7776rabbidv 3443 . . . . . . 7 (𝑏 = 𝑎 → {𝑦𝐷𝑏 < (𝐹𝑦)} = {𝑦𝐷𝑎 < (𝐹𝑦)})
78 fveq2 6905 . . . . . . . . . 10 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7978breq2d 5154 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑎 < (𝐹𝑦) ↔ 𝑎 < (𝐹𝑥)))
8079cbvrabv 3446 . . . . . . . 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 2825 . . . . 5 (𝑏 = 𝑎 → ({𝑦𝐷𝑏 < (𝐹𝑦)} ∈ (𝑆t 𝐷) ↔ {𝑥𝐷𝑎 < (𝐹𝑥)} ∈ (𝑆t 𝐷)))
8483cbvralvw 3236 . . . 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 279 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 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  {crab 3435  Vcvv 3479  wss 3950   cuni 4906   class class class wbr 5142  dom cdm 5684  wf 6556  cfv 6560  (class class class)co 7432  cr 11155  *cxr 11295   < clt 11296  cle 11297  t crest 17466  SAlgcsalg 46328  SMblFncsmblfn 46715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cc 10476  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-card 9980  df-acn 9983  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-ioo 13392  df-ico 13394  df-fl 13833  df-rest 17468  df-salg 46329  df-smblfn 46716
This theorem is referenced by:  issmfgtd  46781  smfpreimagt  46782
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