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Theorem diophrex 39509
Description: Projecting a Diophantine set by removing a coordinate results in a Diophantine set. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Assertion
Ref Expression
diophrex ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑡,𝑁,𝑢   𝑡,𝑆,𝑢
Allowed substitution hints:   𝑀(𝑢,𝑡)

Proof of Theorem diophrex
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2824 . . . . 5 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
21rexbidv 3284 . . . 4 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3 reseq1 5823 . . . . . 6 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
43eqeq2d 2831 . . . . 5 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
54cbvrexvw 3429 . . . 4 (∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
62, 5syl6bb 289 . . 3 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
76cbvabv 2888 . 2 {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8 rexeq 3393 . . . . . 6 (𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)} → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))))
98abbidv 2884 . . . . 5 (𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)} → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))})
109adantl 484 . . . 4 ((((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))})
11 eqeq1 2824 . . . . . . . . . . 11 (𝑑 = 𝑏 → (𝑑 = (𝑒 ↾ (1...𝑀)) ↔ 𝑏 = (𝑒 ↾ (1...𝑀))))
1211anbi1d 631 . . . . . . . . . 10 (𝑑 = 𝑏 → ((𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1312rexbidv 3284 . . . . . . . . 9 (𝑑 = 𝑏 → (∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1413rexab 3666 . . . . . . . 8 (∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏(∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
15 r19.41v 3334 . . . . . . . . . 10 (∃𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
1615exbii 1848 . . . . . . . . 9 (∃𝑏𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏(∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
17 rexcom4 3236 . . . . . . . . . 10 (∃𝑒 ∈ (ℕ0m ℕ)∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
18 anass 471 . . . . . . . . . . . . . 14 (((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
1918exbii 1848 . . . . . . . . . . . . 13 (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
20 vex 3476 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 5875 . . . . . . . . . . . . . 14 (𝑒 ↾ (1...𝑀)) ∈ V
22 reseq1 5823 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
2322eqeq2d 2831 . . . . . . . . . . . . . . 15 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2423anbi2d 630 . . . . . . . . . . . . . 14 (𝑏 = (𝑒 ↾ (1...𝑀)) → (((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))))
2521, 24ceqsexv 3520 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2619, 25bitri 277 . . . . . . . . . . . 12 (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27 ancom 463 . . . . . . . . . . . . 13 (((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0))
28 simpl2 1188 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ𝑁))
29 fzss2 12931 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑀))
30 resabs1 5859 . . . . . . . . . . . . . . . 16 ((1...𝑁) ⊆ (1...𝑀) → ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑒 ↾ (1...𝑁)))
3128, 29, 303syl 18 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) = (𝑒 ↾ (1...𝑁)))
3231eqeq2d 2831 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ↔ 𝑎 = (𝑒 ↾ (1...𝑁))))
3332anbi1d 631 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → ((𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3427, 33syl5bb 285 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3526, 34syl5bb 285 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3635rexbidv 3284 . . . . . . . . . 10 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑒 ∈ (ℕ0m ℕ)∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3717, 36syl5bbr 287 . . . . . . . . 9 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏𝑒 ∈ (ℕ0m ℕ)((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3816, 37syl5bbr 287 . . . . . . . 8 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏(∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
3914, 38syl5bb 285 . . . . . . 7 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → (∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)))
4039abbidv 2884 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)})
41 eldioph3 39500 . . . . . . 7 ((𝑁 ∈ ℕ0𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
42413ad2antl1 1181 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
4340, 42eqeltrd 2911 . . . . 5 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
4443adantr 483 . . . 4 ((((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}) → {𝑎 ∣ ∃𝑏 ∈ {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
4510, 44eqeltrd 2911 . . 3 ((((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) ∧ 𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
46 eldioph3b 39499 . . . . 5 (𝑆 ∈ (Dioph‘𝑀) ↔ (𝑀 ∈ ℕ0 ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}))
4746simprbi 499 . . . 4 (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
48473ad2ant3 1131 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
4945, 48r19.29a 3276 . 2 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
507, 49eqeltrrid 2916 1 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1537  wex 1780  wcel 2114  {cab 2798  wrex 3126  wss 3913  cres 5533  cfv 6331  (class class class)co 7133  m cmap 8384  0cc0 10515  1c1 10516  cn 11616  0cn0 11876  cuz 12222  ...cfz 12876  mzPolycmzp 39456  Diophcdioph 39489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-of 7387  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-dju 9308  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-n0 11877  df-z 11961  df-uz 12223  df-fz 12877  df-hash 13676  df-mzpcl 39457  df-mzp 39458  df-dioph 39490
This theorem is referenced by:  rexrabdioph  39528
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