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Theorem diophren 42801
Description: Change variables in a Diophantine set, using class notation. This allows already proved Diophantine sets to be reused in contexts with more variables. (Contributed by Stefan O'Rear, 16-Oct-2014.) (Revised by Stefan O'Rear, 5-Jun-2015.)
Assertion
Ref Expression
diophren ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
Distinct variable groups:   𝑆,𝑎   𝑀,𝑎   𝑁,𝑎   𝐹,𝑎

Proof of Theorem diophren
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zex 12620 . . . . . 6 ℤ ∈ V
2 difexg 5335 . . . . . 6 (ℤ ∈ V → (ℤ ∖ ℕ) ∈ V)
31, 2ax-mp 5 . . . . 5 (ℤ ∖ ℕ) ∈ V
4 ominf 9292 . . . . . 6 ¬ ω ∈ Fin
5 nnuz 12919 . . . . . . . . . 10 ℕ = (ℤ‘1)
6 0p1e1 12386 . . . . . . . . . . 11 (0 + 1) = 1
76fveq2i 6910 . . . . . . . . . 10 (ℤ‘(0 + 1)) = (ℤ‘1)
85, 7eqtr4i 2766 . . . . . . . . 9 ℕ = (ℤ‘(0 + 1))
98difeq2i 4133 . . . . . . . 8 (ℤ ∖ ℕ) = (ℤ ∖ (ℤ‘(0 + 1)))
10 0z 12622 . . . . . . . . 9 0 ∈ ℤ
11 lzenom 42758 . . . . . . . . 9 (0 ∈ ℤ → (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω)
1210, 11ax-mp 5 . . . . . . . 8 (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω
139, 12eqbrtri 5169 . . . . . . 7 (ℤ ∖ ℕ) ≈ ω
14 enfi 9225 . . . . . . 7 ((ℤ ∖ ℕ) ≈ ω → ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin))
1513, 14ax-mp 5 . . . . . 6 ((ℤ ∖ ℕ) ∈ Fin ↔ ω ∈ Fin)
164, 15mtbir 323 . . . . 5 ¬ (ℤ ∖ ℕ) ∈ Fin
17 disjdifr 4479 . . . . 5 ((ℤ ∖ ℕ) ∩ ℕ) = ∅
183, 16, 17eldioph4b 42799 . . . 4 (𝑆 ∈ (Dioph‘𝑁) ↔ (𝑁 ∈ ℕ0 ∧ ∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}))
19 simpr 484 . . . . . . . . . . . 12 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) → 𝑎 ∈ (ℕ0m (1...𝑀)))
20 simp-4r 784 . . . . . . . . . . . 12 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) → 𝐹:(1...𝑁)⟶(1...𝑀))
21 ovex 7464 . . . . . . . . . . . . 13 (1...𝑁) ∈ V
2221mapco2 42703 . . . . . . . . . . . 12 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎𝐹) ∈ (ℕ0m (1...𝑁)))
2319, 20, 22syl2anc 584 . . . . . . . . . . 11 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) → (𝑎𝐹) ∈ (ℕ0m (1...𝑁)))
24 uneq1 4171 . . . . . . . . . . . . . 14 (𝑐 = (𝑎𝐹) → (𝑐𝑑) = ((𝑎𝐹) ∪ 𝑑))
2524fveqeq2d 6915 . . . . . . . . . . . . 13 (𝑐 = (𝑎𝐹) → ((𝑏‘(𝑐𝑑)) = 0 ↔ (𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2625rexbidv 3177 . . . . . . . . . . . 12 (𝑐 = (𝑎𝐹) → (∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2726elrab3 3696 . . . . . . . . . . 11 ((𝑎𝐹) ∈ (ℕ0m (1...𝑁)) → ((𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2823, 27syl 17 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) → ((𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
29 simp-5r 786 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
30 simplr 769 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝑎 ∈ (ℕ0m (1...𝑀)))
31 simpr 484 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)))
32 coundi 6269 . . . . . . . . . . . . . . . 16 ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))))
33 coundir 6270 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ (𝑑𝐹))
34 elmapi 8888 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
35343ad2ant3 1134 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
36 simp1 1135 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
37 incom 4217 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖ ℕ))
38 fz1ssnn 13592 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...𝑀) ⊆ ℕ
39 disjdif 4478 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℕ ∩ (ℤ ∖ ℕ)) = ∅
40 ssdisj 4466 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1...𝑀) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅)
4138, 39, 40mp2an 692 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅
4237, 41eqtri 2763 . . . . . . . . . . . . . . . . . . . . . 22 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅
4342a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅)
44 coeq0i 42741 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑:(ℤ ∖ ℕ)⟶ℕ0𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) → (𝑑𝐹) = ∅)
4535, 36, 43, 44syl3anc 1370 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑑𝐹) = ∅)
4645uneq2d 4178 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ (𝑑𝐹)) = ((𝑎𝐹) ∪ ∅))
4733, 46eqtrid 2787 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ ∅))
48 un0 4400 . . . . . . . . . . . . . . . . . 18 ((𝑎𝐹) ∪ ∅) = (𝑎𝐹)
4947, 48eqtrdi 2791 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = (𝑎𝐹))
50 coundir 6270 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))))
51 elmapi 8888 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (ℕ0m (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0)
52513ad2ant2 1133 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0)
53 f1oi 6887 . . . . . . . . . . . . . . . . . . . . . . 23 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ)
54 f1of 6849 . . . . . . . . . . . . . . . . . . . . . . 23 (( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
5553, 54ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ)
56 coeq0i 42741 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5755, 41, 56mp3an23 1452 . . . . . . . . . . . . . . . . . . . . 21 (𝑎:(1...𝑀)⟶ℕ0 → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5852, 57syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
59 coires1 6286 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = (𝑑 ↾ (ℤ ∖ ℕ))
60 ffn 6737 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑:(ℤ ∖ ℕ)⟶ℕ0𝑑 Fn (ℤ ∖ ℕ))
61 fnresdm 6688 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 Fn (ℤ ∖ ℕ) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6234, 60, 613syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6359, 62eqtrid 2787 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
64633ad2ant3 1134 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
6558, 64uneq12d 4179 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ)))) = (∅ ∪ 𝑑))
6650, 65eqtrid 2787 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = (∅ ∪ 𝑑))
67 uncom 4168 . . . . . . . . . . . . . . . . . . 19 (∅ ∪ 𝑑) = (𝑑 ∪ ∅)
68 un0 4400 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∪ ∅) = 𝑑
6967, 68eqtri 2763 . . . . . . . . . . . . . . . . . 18 (∅ ∪ 𝑑) = 𝑑
7066, 69eqtrdi 2791 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
7149, 70uneq12d 4179 . . . . . . . . . . . . . . . 16 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝐹) ∪ 𝑑))
7232, 71eqtr2id 2788 . . . . . . . . . . . . . . 15 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ 𝑑) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
7329, 30, 31, 72syl3anc 1370 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ 𝑑) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
7473fveq2d 6911 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑏‘((𝑎𝐹) ∪ 𝑑)) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
75 nn0ssz 12634 . . . . . . . . . . . . . . . . 17 0 ⊆ ℤ
76 mapss 8928 . . . . . . . . . . . . . . . . 17 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
771, 75, 76mp2an 692 . . . . . . . . . . . . . . . 16 (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀)))
7841reseq2i 5997 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾ ∅)
79 res0 6004 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ∅) = ∅
8078, 79eqtri 2763 . . . . . . . . . . . . . . . . . 18 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8141reseq2i 5997 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ∅)
82 res0 6004 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ∅) = ∅
8381, 82eqtri 2763 . . . . . . . . . . . . . . . . . 18 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8480, 83eqtr4i 2766 . . . . . . . . . . . . . . . . 17 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))
85 elmapresaun 8919 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) → (𝑎𝑑) ∈ (ℕ0m ((1...𝑀) ∪ (ℤ ∖ ℕ))))
86 uncom 4168 . . . . . . . . . . . . . . . . . . 19 ((1...𝑀) ∪ (ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀))
8786oveq2i 7442 . . . . . . . . . . . . . . . . . 18 (ℕ0m ((1...𝑀) ∪ (ℤ ∖ ℕ))) = (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀)))
8885, 87eleqtrdi 2849 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) ∧ (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))) → (𝑎𝑑) ∈ (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
8984, 88mp3an3 1449 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9077, 89sselid 3993 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9190adantll 714 . . . . . . . . . . . . . 14 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
92 coeq1 5871 . . . . . . . . . . . . . . . 16 (𝑒 = (𝑎𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
9392fveq2d 6911 . . . . . . . . . . . . . . 15 (𝑒 = (𝑎𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
94 eqid 2735 . . . . . . . . . . . . . . 15 (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) = (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
95 fvex 6920 . . . . . . . . . . . . . . 15 (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) ∈ V
9693, 94, 95fvmpt 7016 . . . . . . . . . . . . . 14 ((𝑎𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) → ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
9791, 96syl 17 . . . . . . . . . . . . 13 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
9874, 97eqtr4d 2778 . . . . . . . . . . . 12 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑏‘((𝑎𝐹) ∪ 𝑑)) = ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)))
9998eqeq1d 2737 . . . . . . . . . . 11 ((((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0 ↔ ((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0))
10099rexbidva 3175 . . . . . . . . . 10 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) → (∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0))
10128, 100bitrd 279 . . . . . . . . 9 (((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) ∧ 𝑎 ∈ (ℕ0m (1...𝑀))) → ((𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} ↔ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0))
102101rabbidva 3440 . . . . . . . 8 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}} = {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0})
103 simplll 775 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈ ℕ0)
104 ovex 7464 . . . . . . . . . . . 12 (1...𝑀) ∈ V
1053, 104unex 7763 . . . . . . . . . . 11 ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V
106105a1i 11 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → ((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V)
107 simpr 484 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁))))
10855a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ))
109 id 22 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → 𝐹:(1...𝑁)⟶(1...𝑀))
110 incom 4217 . . . . . . . . . . . . . . 15 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖ ℕ))
111 fz1ssnn 13592 . . . . . . . . . . . . . . . 16 (1...𝑁) ⊆ ℕ
112 ssdisj 4466 . . . . . . . . . . . . . . . 16 (((1...𝑁) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅)
113111, 39, 112mp2an 692 . . . . . . . . . . . . . . 15 ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅
114110, 113eqtri 2763 . . . . . . . . . . . . . 14 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅
115114a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅)
116 fun 6771 . . . . . . . . . . . . 13 (((( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
117108, 109, 115, 116syl21anc 838 . . . . . . . . . . . 12 (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
118 uncom 4168 . . . . . . . . . . . . 13 (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))
119118feq1i 6728 . . . . . . . . . . . 12 ((( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)) ↔ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
120117, 119sylib 218 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
121120ad3antlr 731 . . . . . . . . . 10 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
122 mzprename 42737 . . . . . . . . . 10 ((((ℤ ∖ ℕ) ∪ (1...𝑀)) ∈ V ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁))) ∧ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀))) → (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀))))
123106, 107, 121, 122syl3anc 1370 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀))))
1243, 16, 17eldioph4i 42800 . . . . . . . . 9 ((𝑀 ∈ ℕ0 ∧ (𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))) ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑀)))) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0} ∈ (Dioph‘𝑀))
125103, 123, 124syl2anc 584 . . . . . . . 8 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))((𝑒 ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ↦ (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))‘(𝑎𝑑)) = 0} ∈ (Dioph‘𝑀))
126102, 125eqeltrd 2839 . . . . . . 7 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}} ∈ (Dioph‘𝑀))
127 eleq2 2828 . . . . . . . . 9 (𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → ((𝑎𝐹) ∈ 𝑆 ↔ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}))
128127rabbidv 3441 . . . . . . . 8 (𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} = {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}})
129128eleq1d 2824 . . . . . . 7 (𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → ({𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀) ↔ {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}} ∈ (Dioph‘𝑀)))
130126, 129syl5ibrcom 247 . . . . . 6 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → (𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
131130rexlimdva 3153 . . . . 5 (((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) → (∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0} → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
132131expimpd 453 . . . 4 ((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → ((𝑁 ∈ ℕ0 ∧ ∃𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))𝑆 = {𝑐 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0}) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
13318, 132biimtrid 242 . . 3 ((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑆 ∈ (Dioph‘𝑁) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀)))
134133impcom 407 . 2 ((𝑆 ∈ (Dioph‘𝑁) ∧ (𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀))) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
1351343impb 1114 1 ((𝑆 ∈ (Dioph‘𝑁) ∧ 𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) → {𝑎 ∈ (ℕ0m (1...𝑀)) ∣ (𝑎𝐹) ∈ 𝑆} ∈ (Dioph‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339   class class class wbr 5148  cmpt 5231   I cid 5582  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  ωcom 7887  m cmap 8865  cen 8981  Fincfn 8984  0cc0 11153  1c1 11154   + caddc 11156  cn 12264  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  mzPolycmzp 42710  Diophcdioph 42743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-hash 14367  df-mzpcl 42711  df-mzp 42712  df-dioph 42744
This theorem is referenced by:  rabrenfdioph  42802
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