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Theorem diophrex 42762
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 2738 . . . . 5 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
21rexbidv 3176 . . . 4 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3 reseq1 5993 . . . . . 6 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
43eqeq2d 2745 . . . . 5 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
54cbvrexvw 3235 . . . 4 (∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁)))
62, 5bitrdi 287 . . 3 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))))
76cbvabv 2809 . 2 {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8 rexeq 3319 . . . . . 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 2805 . . . . 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 481 . . . 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 2738 . . . . . . . . . . 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 3176 . . . . . . . . 9 (𝑑 = 𝑏 → (∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1413rexab 3702 . . . . . . . 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 3186 . . . . . . . . . 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 1844 . . . . . . . . 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 3285 . . . . . . . . . 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 468 . . . . . . . . . . . . . 14 (((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ (𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
1918exbii 1844 . . . . . . . . . . . . 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 3481 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 6048 . . . . . . . . . . . . . 14 (𝑒 ↾ (1...𝑀)) ∈ V
22 reseq1 5993 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
2322eqeq2d 2745 . . . . . . . . . . . . . . 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 3529 . . . . . . . . . . . . 13 (∃𝑏(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ ((𝑐𝑒) = 0 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
2619, 25bitri 275 . . . . . . . . . . . 12 (∃𝑏((𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))))
27 ancom 460 . . . . . . . . . . . . 13 (((𝑐𝑒) = 0 ∧ 𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁))) ↔ (𝑎 = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0))
28 simpl2 1191 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ𝑁))
29 fzss2 13600 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑀))
30 resabs1 6026 . . . . . . . . . . . . . . . 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 2745 . . . . . . . . . . . . . 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, 33bitrid 283 . . . . . . . . . . . 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, 34bitrid 283 . . . . . . . . . . 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 3176 . . . . . . . . . 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, 36bitr3id 285 . . . . . . . . 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, 37bitr3id 285 . . . . . . . 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, 38bitrid 283 . . . . . . 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 2805 . . . . . 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 42753 . . . . . . 7 ((𝑁 ∈ ℕ0𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
42413ad2antl1 1184 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
4340, 42eqeltrd 2838 . . . . 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 480 . . . 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 2838 . . 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 42752 . . . . 5 (𝑆 ∈ (Dioph‘𝑀) ↔ (𝑀 ∈ ℕ0 ∧ ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)}))
4746simprbi 496 . . . 4 (𝑆 ∈ (Dioph‘𝑀) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
48473ad2ant3 1134 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
4945, 48r19.29a 3159 . 2 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
507, 49eqeltrrid 2843 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 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wrex 3067  wss 3962  cres 5690  cfv 6562  (class class class)co 7430  m cmap 8864  0cc0 11152  1c1 11153  cn 12263  0cn0 12523  cuz 12875  ...cfz 13543  mzPolycmzp 42709  Diophcdioph 42742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366  df-mzpcl 42710  df-mzp 42711  df-dioph 42743
This theorem is referenced by:  rexrabdioph  42781
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