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Theorem diophrex 42786
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 2741 . . . . 5 (𝑎 = 𝑡 → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑏 ↾ (1...𝑁))))
21rexbidv 3179 . . . 4 (𝑎 = 𝑡 → (∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏𝑆 𝑡 = (𝑏 ↾ (1...𝑁))))
3 reseq1 5991 . . . . . 6 (𝑏 = 𝑢 → (𝑏 ↾ (1...𝑁)) = (𝑢 ↾ (1...𝑁)))
43eqeq2d 2748 . . . . 5 (𝑏 = 𝑢 → (𝑡 = (𝑏 ↾ (1...𝑁)) ↔ 𝑡 = (𝑢 ↾ (1...𝑁))))
54cbvrexvw 3238 . . . 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 2812 . 2 {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑡 ∣ ∃𝑢𝑆 𝑡 = (𝑢 ↾ (1...𝑁))}
8 rexeq 3322 . . . . . 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 2808 . . . . 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 2741 . . . . . . . . . . 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 3179 . . . . . . . . 9 (𝑑 = 𝑏 → (∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0) ↔ ∃𝑒 ∈ (ℕ0m ℕ)(𝑏 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)))
1413rexab 3700 . . . . . . . 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 3189 . . . . . . . . . 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 3288 . . . . . . . . . 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 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 3484 . . . . . . . . . . . . . . 15 𝑒 ∈ V
2120resex 6047 . . . . . . . . . . . . . 14 (𝑒 ↾ (1...𝑀)) ∈ V
22 reseq1 5991 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑒 ↾ (1...𝑀)) → (𝑏 ↾ (1...𝑁)) = ((𝑒 ↾ (1...𝑀)) ↾ (1...𝑁)))
2322eqeq2d 2748 . . . . . . . . . . . . . . 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 3532 . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → 𝑀 ∈ (ℤ𝑁))
29 fzss2 13604 . . . . . . . . . . . . . . . 16 (𝑀 ∈ (ℤ𝑁) → (1...𝑁) ⊆ (1...𝑀))
30 resabs1 6024 . . . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . . 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 3179 . . . . . . . . . 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 2808 . . . . . 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 42777 . . . . . . 7 ((𝑁 ∈ ℕ0𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
42413ad2antl1 1186 . . . . . 6 (((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) ∧ 𝑐 ∈ (mzPoly‘ℕ)) → {𝑎 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑎 = (𝑒 ↾ (1...𝑁)) ∧ (𝑐𝑒) = 0)} ∈ (Dioph‘𝑁))
4340, 42eqeltrd 2841 . . . . 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 2841 . . 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 42776 . . . . 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 1136 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → ∃𝑐 ∈ (mzPoly‘ℕ)𝑆 = {𝑑 ∣ ∃𝑒 ∈ (ℕ0m ℕ)(𝑑 = (𝑒 ↾ (1...𝑀)) ∧ (𝑐𝑒) = 0)})
4945, 48r19.29a 3162 . 2 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ 𝑆 ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏𝑆 𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
507, 49eqeltrrid 2846 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 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wrex 3070  wss 3951  cres 5687  cfv 6561  (class class class)co 7431  m cmap 8866  0cc0 11155  1c1 11156  cn 12266  0cn0 12526  cuz 12878  ...cfz 13547  mzPolycmzp 42733  Diophcdioph 42766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-hash 14370  df-mzpcl 42734  df-mzp 42735  df-dioph 42767
This theorem is referenced by:  rexrabdioph  42805
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