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Theorem diophren 42006
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 12563 . . . . . 6 ℤ ∈ V
2 difexg 5317 . . . . . 6 (ℤ ∈ V → (ℤ ∖ ℕ) ∈ V)
31, 2ax-mp 5 . . . . 5 (ℤ ∖ ℕ) ∈ V
4 ominf 9253 . . . . . 6 ¬ ω ∈ Fin
5 nnuz 12861 . . . . . . . . . 10 ℕ = (ℤ‘1)
6 0p1e1 12330 . . . . . . . . . . 11 (0 + 1) = 1
76fveq2i 6884 . . . . . . . . . 10 (ℤ‘(0 + 1)) = (ℤ‘1)
85, 7eqtr4i 2755 . . . . . . . . 9 ℕ = (ℤ‘(0 + 1))
98difeq2i 4111 . . . . . . . 8 (ℤ ∖ ℕ) = (ℤ ∖ (ℤ‘(0 + 1)))
10 0z 12565 . . . . . . . . 9 0 ∈ ℤ
11 lzenom 41963 . . . . . . . . 9 (0 ∈ ℤ → (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω)
1210, 11ax-mp 5 . . . . . . . 8 (ℤ ∖ (ℤ‘(0 + 1))) ≈ ω
139, 12eqbrtri 5159 . . . . . . 7 (ℤ ∖ ℕ) ≈ ω
14 enfi 9185 . . . . . . 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 4464 . . . . 5 ((ℤ ∖ ℕ) ∩ ℕ) = ∅
183, 16, 17eldioph4b 42004 . . . 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 781 . . . . . . . . . . . 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 7434 . . . . . . . . . . . . 13 (1...𝑁) ∈ V
2221mapco2 41908 . . . . . . . . . . . 12 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝐹:(1...𝑁)⟶(1...𝑀)) → (𝑎𝐹) ∈ (ℕ0m (1...𝑁)))
2319, 20, 22syl2anc 583 . . . . . . . . . . 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 4148 . . . . . . . . . . . . . 14 (𝑐 = (𝑎𝐹) → (𝑐𝑑) = ((𝑎𝐹) ∪ 𝑑))
2524fveqeq2d 6889 . . . . . . . . . . . . 13 (𝑐 = (𝑎𝐹) → ((𝑏‘(𝑐𝑑)) = 0 ↔ (𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2625rexbidv 3170 . . . . . . . . . . . 12 (𝑐 = (𝑎𝐹) → (∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘(𝑐𝑑)) = 0 ↔ ∃𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))(𝑏‘((𝑎𝐹) ∪ 𝑑)) = 0))
2726elrab3 3676 . . . . . . . . . . 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 783 . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . 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 6236 . . . . . . . . . . . . . . . 16 ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))))
33 coundir 6237 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ (𝑑𝐹))
34 elmapi 8838 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
35343ad2ant3 1132 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝑑:(ℤ ∖ ℕ)⟶ℕ0)
36 simp1 1133 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝐹:(1...𝑁)⟶(1...𝑀))
37 incom 4193 . . . . . . . . . . . . . . . . . . . . . . 23 ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ((1...𝑀) ∩ (ℤ ∖ ℕ))
38 fz1ssnn 13528 . . . . . . . . . . . . . . . . . . . . . . . 24 (1...𝑀) ⊆ ℕ
39 disjdif 4463 . . . . . . . . . . . . . . . . . . . . . . . 24 (ℕ ∩ (ℤ ∖ ℕ)) = ∅
40 ssdisj 4451 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1...𝑀) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅)
4138, 39, 40mp2an 689 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅
4237, 41eqtri 2752 . . . . . . . . . . . . . . . . . . . . . 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 41946 . . . . . . . . . . . . . . . . . . . . 21 ((𝑑:(ℤ ∖ ℕ)⟶ℕ0𝐹:(1...𝑁)⟶(1...𝑀) ∧ ((ℤ ∖ ℕ) ∩ (1...𝑀)) = ∅) → (𝑑𝐹) = ∅)
4535, 36, 43, 44syl3anc 1368 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑑𝐹) = ∅)
4645uneq2d 4155 . . . . . . . . . . . . . . . . . . 19 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝐹) ∪ (𝑑𝐹)) = ((𝑎𝐹) ∪ ∅))
4733, 46eqtrid 2776 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = ((𝑎𝐹) ∪ ∅))
48 un0 4382 . . . . . . . . . . . . . . . . . 18 ((𝑎𝐹) ∪ ∅) = (𝑎𝐹)
4947, 48eqtrdi 2780 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ 𝐹) = (𝑎𝐹))
50 coundir 6237 . . . . . . . . . . . . . . . . . . 19 ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = ((𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) ∪ (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))))
51 elmapi 8838 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 ∈ (ℕ0m (1...𝑀)) → 𝑎:(1...𝑀)⟶ℕ0)
52513ad2ant2 1131 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → 𝑎:(1...𝑀)⟶ℕ0)
53 f1oi 6861 . . . . . . . . . . . . . . . . . . . . . . 23 ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)–1-1-onto→(ℤ ∖ ℕ)
54 f1of 6823 . . . . . . . . . . . . . . . . . . . . . . 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 41946 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎:(1...𝑀)⟶ℕ0 ∧ ( I ↾ (ℤ ∖ ℕ)):(ℤ ∖ ℕ)⟶(ℤ ∖ ℕ) ∧ ((1...𝑀) ∩ (ℤ ∖ ℕ)) = ∅) → (𝑎 ∘ ( I ↾ (ℤ ∖ ℕ))) = ∅)
5755, 41, 56mp3an23 1449 . . . . . . . . . . . . . . . . . . . . 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 6253 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = (𝑑 ↾ (ℤ ∖ ℕ))
60 ffn 6707 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑:(ℤ ∖ ℕ)⟶ℕ0𝑑 Fn (ℤ ∖ ℕ))
61 fnresdm 6659 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑑 Fn (ℤ ∖ ℕ) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6234, 60, 613syl 18 . . . . . . . . . . . . . . . . . . . . . 22 (𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) → (𝑑 ↾ (ℤ ∖ ℕ)) = 𝑑)
6359, 62eqtrid 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ∈ (ℕ0m (ℤ ∖ ℕ)) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
64633ad2ant3 1132 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑑 ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
6558, 64uneq12d 4156 . . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . . 18 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = (∅ ∪ 𝑑))
67 uncom 4145 . . . . . . . . . . . . . . . . . . 19 (∅ ∪ 𝑑) = (𝑑 ∪ ∅)
68 un0 4382 . . . . . . . . . . . . . . . . . . 19 (𝑑 ∪ ∅) = 𝑑
6967, 68eqtri 2752 . . . . . . . . . . . . . . . . . 18 (∅ ∪ 𝑑) = 𝑑
7066, 69eqtrdi 2780 . . . . . . . . . . . . . . . . 17 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ))) = 𝑑)
7149, 70uneq12d 4156 . . . . . . . . . . . . . . . 16 ((𝐹:(1...𝑁)⟶(1...𝑀) ∧ 𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (((𝑎𝑑) ∘ 𝐹) ∪ ((𝑎𝑑) ∘ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝐹) ∪ 𝑑))
7232, 71eqtr2id 2777 . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . 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 6885 . . . . . . . . . . . . 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 12577 . . . . . . . . . . . . . . . . 17 0 ⊆ ℤ
76 mapss 8878 . . . . . . . . . . . . . . . . 17 ((ℤ ∈ V ∧ ℕ0 ⊆ ℤ) → (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
771, 75, 76mp2an 689 . . . . . . . . . . . . . . . 16 (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))) ⊆ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀)))
7841reseq2i 5968 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑎 ↾ ∅)
79 res0 5975 . . . . . . . . . . . . . . . . . . 19 (𝑎 ↾ ∅) = ∅
8078, 79eqtri 2752 . . . . . . . . . . . . . . . . . 18 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8141reseq2i 5968 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ∅)
82 res0 5975 . . . . . . . . . . . . . . . . . . 19 (𝑑 ↾ ∅) = ∅
8381, 82eqtri 2752 . . . . . . . . . . . . . . . . . 18 (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = ∅
8480, 83eqtr4i 2755 . . . . . . . . . . . . . . . . 17 (𝑎 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ))) = (𝑑 ↾ ((1...𝑀) ∩ (ℤ ∖ ℕ)))
85 elmapresaun 8869 . . . . . . . . . . . . . . . . . 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 4145 . . . . . . . . . . . . . . . . . . 19 ((1...𝑀) ∪ (ℤ ∖ ℕ)) = ((ℤ ∖ ℕ) ∪ (1...𝑀))
8786oveq2i 7412 . . . . . . . . . . . . . . . . . 18 (ℕ0m ((1...𝑀) ∪ (ℤ ∖ ℕ))) = (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀)))
8885, 87eleqtrdi 2835 . . . . . . . . . . . . . . . . 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 1446 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℕ0m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9077, 89sselid 3972 . . . . . . . . . . . . . . 15 ((𝑎 ∈ (ℕ0m (1...𝑀)) ∧ 𝑑 ∈ (ℕ0m (ℤ ∖ ℕ))) → (𝑎𝑑) ∈ (ℤ ↑m ((ℤ ∖ ℕ) ∪ (1...𝑀))))
9190adantll 711 . . . . . . . . . . . . . 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 5847 . . . . . . . . . . . . . . . 16 (𝑒 = (𝑎𝑑) → (𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))) = ((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))))
9392fveq2d 6885 . . . . . . . . . . . . . . 15 (𝑒 = (𝑎𝑑) → (𝑏‘(𝑒 ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) = (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))))
94 eqid 2724 . . . . . . . . . . . . . . 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 6894 . . . . . . . . . . . . . . 15 (𝑏‘((𝑎𝑑) ∘ (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))))) ∈ V
9693, 94, 95fvmpt 6988 . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . 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 2726 . . . . . . . . . . 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 3168 . . . . . . . . . 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 3431 . . . . . . . 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 772 . . . . . . . . 9 ((((𝑀 ∈ ℕ0𝐹:(1...𝑁)⟶(1...𝑀)) ∧ 𝑁 ∈ ℕ0) ∧ 𝑏 ∈ (mzPoly‘((ℤ ∖ ℕ) ∪ (1...𝑁)))) → 𝑀 ∈ ℕ0)
104 ovex 7434 . . . . . . . . . . . 12 (1...𝑀) ∈ V
1053, 104unex 7726 . . . . . . . . . . 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 4193 . . . . . . . . . . . . . . 15 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ((1...𝑁) ∩ (ℤ ∖ ℕ))
111 fz1ssnn 13528 . . . . . . . . . . . . . . . 16 (1...𝑁) ⊆ ℕ
112 ssdisj 4451 . . . . . . . . . . . . . . . 16 (((1...𝑁) ⊆ ℕ ∧ (ℕ ∩ (ℤ ∖ ℕ)) = ∅) → ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅)
113111, 39, 112mp2an 689 . . . . . . . . . . . . . . 15 ((1...𝑁) ∩ (ℤ ∖ ℕ)) = ∅
114110, 113eqtri 2752 . . . . . . . . . . . . . 14 ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅
115114a1i 11 . . . . . . . . . . . . 13 (𝐹:(1...𝑁)⟶(1...𝑀) → ((ℤ ∖ ℕ) ∩ (1...𝑁)) = ∅)
116 fun 6743 . . . . . . . . . . . . 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 835 . . . . . . . . . . . 12 (𝐹:(1...𝑁)⟶(1...𝑀) → (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
118 uncom 4145 . . . . . . . . . . . . 13 (( I ↾ (ℤ ∖ ℕ)) ∪ 𝐹) = (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ)))
119118feq1i 6698 . . . . . . . . . . . 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 217 . . . . . . . . . . 11 (𝐹:(1...𝑁)⟶(1...𝑀) → (𝐹 ∪ ( I ↾ (ℤ ∖ ℕ))):((ℤ ∖ ℕ) ∪ (1...𝑁))⟶((ℤ ∖ ℕ) ∪ (1...𝑀)))
121120ad3antlr 728 . . . . . . . . . 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 41942 . . . . . . . . . 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 1368 . . . . . . . . 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 42005 . . . . . . . . 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 583 . . . . . . . 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 2825 . . . . . . 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 2814 . . . . . . . . 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 3432 . . . . . . . 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 2810 . . . . . . 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 246 . . . . . 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 3147 . . . . 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 241 . . 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 1112 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 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wrex 3062  {crab 3424  Vcvv 3466  cdif 3937  cun 3938  cin 3939  wss 3940  c0 4314   class class class wbr 5138  cmpt 5221   I cid 5563  cres 5668  ccom 5670   Fn wfn 6528  wf 6529  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  ωcom 7848  m cmap 8815  cen 8931  Fincfn 8934  0cc0 11105  1c1 11106   + caddc 11108  cn 12208  0cn0 12468  cz 12554  cuz 12818  ...cfz 13480  mzPolycmzp 41915  Diophcdioph 41948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-mzpcl 41916  df-mzp 41917  df-dioph 41949
This theorem is referenced by:  rabrenfdioph  42007
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