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Theorem rexrabdioph 36242
Description: Diophantine set builder for existential quantification. (Contributed by Stefan O'Rear, 10-Oct-2014.)
Hypotheses
Ref Expression
rexrabdioph.1 𝑀 = (𝑁 + 1)
rexrabdioph.2 (𝑣 = (𝑡𝑀) → (𝜓𝜒))
rexrabdioph.3 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))
Assertion
Ref Expression
rexrabdioph ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁))
Distinct variable groups:   𝑡,𝑁,𝑢,𝑣   𝑡,𝑀,𝑢,𝑣   𝜑,𝑢,𝑣   𝜓,𝑡   𝜒,𝑣
Allowed substitution hints:   𝜑(𝑡)   𝜓(𝑣,𝑢)   𝜒(𝑢,𝑡)

Proof of Theorem rexrabdioph
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rab 2809 . . . . . 6 {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓} = {𝑎 ∣ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)}
2 dfsbcq 3308 . . . . . . . . . . 11 (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓[𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
32cbvrexv 3052 . . . . . . . . . 10 (∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
43anbi2i 725 . . . . . . . . 9 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
5 r19.42v 2977 . . . . . . . . 9 (∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
64, 5bitr4i 265 . . . . . . . 8 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
7 simpll 785 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑁 ∈ ℕ0)
8 simpr 475 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
9 simplr 787 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 ∈ ℕ0)
10 rexrabdioph.1 . . . . . . . . . . . . . . 15 𝑀 = (𝑁 + 1)
1110mapfzcons 36163 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
127, 8, 9, 11syl3anc 1317 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1312adantrr 748 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1410mapfzcons2 36166 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
158, 9, 14syl2anc 690 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
1615eqcomd 2520 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
1710mapfzcons1 36164 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1817adantl 480 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1918eqcomd 2520 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2019sbceq1d 3311 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ([𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2116, 20sbceqbid 3313 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2221biimpd 217 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2322impr 646 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → [((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓)
2419adantrr 748 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25 fveq1 5985 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26 reseq1 5202 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2726sbceq1d 3311 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2825, 27sbceqbid 3313 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2926eqeq2d 2524 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
3028, 29anbi12d 742 . . . . . . . . . . . . 13 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ([((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))))
3130rspcev 3186 . . . . . . . . . . . 12 (((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)) ∧ ([((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))) → ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁))))
3213, 23, 24, 31syl12anc 1315 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁))))
3332ex 448 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) → ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) → ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))))
3433rexlimdva 2917 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) → ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))))
35 elmapi 7639 . . . . . . . . . . . . . 14 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ0)
36 nn0p1nn 11085 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3710, 36syl5eqel 2596 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
38 elfz1end 12106 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
3937, 38sylib 206 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
40 ffvelrn 6148 . . . . . . . . . . . . . 14 ((𝑏:(1...𝑀)⟶ℕ0𝑀 ∈ (1...𝑀)) → (𝑏𝑀) ∈ ℕ0)
4135, 39, 40syl2anr 493 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) → (𝑏𝑀) ∈ ℕ0)
4241adantr 479 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏𝑀) ∈ ℕ0)
43 simprr 791 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
4410mapfzcons1cl 36165 . . . . . . . . . . . . . 14 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4544ad2antlr 758 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4643, 45eqeltrd 2592 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
47 simprl 789 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)
48 dfsbcq 3308 . . . . . . . . . . . . . . 15 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4948sbcbidv 3361 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
5049ad2antll 760 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → ([(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
5147, 50mpbird 245 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)
52 dfsbcq 3308 . . . . . . . . . . . . . 14 (𝑐 = (𝑏𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
5352anbi2d 735 . . . . . . . . . . . . 13 (𝑐 = (𝑏𝑀) → ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
5453rspcev 3186 . . . . . . . . . . . 12 (((𝑏𝑀) ∈ ℕ0 ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)) → ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
5542, 46, 51, 54syl12anc 1315 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
5655ex 448 . . . . . . . . . 10 ((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) → (([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁))) → ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)))
5756rexlimdva 2917 . . . . . . . . 9 (𝑁 ∈ ℕ0 → (∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁))) → ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)))
5834, 57impbid 200 . . . . . . . 8 (𝑁 ∈ ℕ0 → (∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))))
596, 58syl5bb 270 . . . . . . 7 (𝑁 ∈ ℕ0 → ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))))
6059abbidv 2632 . . . . . 6 (𝑁 ∈ ℕ0 → {𝑎 ∣ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))})
611, 60syl5eq 2560 . . . . 5 (𝑁 ∈ ℕ0 → {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))})
62 nfcv 2655 . . . . . 6 𝑢(ℕ0𝑚 (1...𝑁))
63 nfcv 2655 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
64 nfv 1796 . . . . . 6 𝑎𝑣 ∈ ℕ0 𝜓
65 nfcv 2655 . . . . . . 7 𝑢0
66 nfcv 2655 . . . . . . . 8 𝑢𝑏
67 nfsbc1v 3326 . . . . . . . 8 𝑢[𝑎 / 𝑢]𝜓
6866, 67nfsbc 3328 . . . . . . 7 𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
6965, 68nfrex 2894 . . . . . 6 𝑢𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
70 sbceq1a 3317 . . . . . . . 8 (𝑢 = 𝑎 → (𝜓[𝑎 / 𝑢]𝜓))
7170rexbidv 2938 . . . . . . 7 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓))
72 nfv 1796 . . . . . . . 8 𝑏[𝑎 / 𝑢]𝜓
73 nfsbc1v 3326 . . . . . . . 8 𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
74 sbceq1a 3317 . . . . . . . 8 (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓[𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7572, 73, 74cbvrex 3048 . . . . . . 7 (∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
7671, 75syl6bb 274 . . . . . 6 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7762, 63, 64, 69, 76cbvrab 3075 . . . . 5 {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
78 fveq1 5985 . . . . . . . 8 (𝑡 = 𝑏 → (𝑡𝑀) = (𝑏𝑀))
79 reseq1 5202 . . . . . . . . 9 (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
8079sbceq1d 3311 . . . . . . . 8 (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8178, 80sbceqbid 3313 . . . . . . 7 (𝑡 = 𝑏 → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8281rexrab 3241 . . . . . 6 (∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁))))
8382abbii 2630 . . . . 5 {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ (ℕ0𝑚 (1...𝑀))([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))}
8461, 77, 833eqtr4g 2573 . . . 4 (𝑁 ∈ ℕ0 → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁))})
85 fvex 5996 . . . . . . . . 9 (𝑡𝑀) ∈ V
86 vex 3080 . . . . . . . . . 10 𝑡 ∈ V
8786resex 5254 . . . . . . . . 9 (𝑡 ↾ (1...𝑁)) ∈ V
88 rexrabdioph.2 . . . . . . . . . 10 (𝑣 = (𝑡𝑀) → (𝜓𝜒))
89 rexrabdioph.3 . . . . . . . . . 10 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))
9088, 89sylan9bb 731 . . . . . . . . 9 ((𝑣 = (𝑡𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓𝜑))
9185, 87, 90sbc2ie 3376 . . . . . . . 8 ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑)
9291a1i 11 . . . . . . 7 (𝑡 ∈ (ℕ0𝑚 (1...𝑀)) → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑))
9392rabbiia 3065 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}
9493rexeqi 3024 . . . . 5 (∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁)))
9594abbii 2630 . . . 4 {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))}
9684, 95syl6eq 2564 . . 3 (𝑁 ∈ ℕ0 → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
9796adantr 479 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
98 simpl 471 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ0)
99 nn0z 11139 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
100 uzid 11438 . . . . . 6 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
101 peano2uz 11477 . . . . . 6 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
10299, 100, 1013syl 18 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (ℤ𝑁))
10310, 102syl5eqel 2596 . . . 4 (𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁))
104103adantr 479 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ𝑁))
105 simpr 475 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
106 diophrex 36223 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
10798, 104, 105, 106syl3anc 1317 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
10897, 107eqeltrd 2592 1 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1938  {cab 2500  wrex 2801  {crab 2804  [wsbc 3306  cun 3442  {csn 4028  cop 4034  cres 4934  wf 5685  cfv 5689  (class class class)co 6425  𝑚 cmap 7618  1c1 9690   + caddc 9692  cn 10773  0cn0 11045  cz 11116  cuz 11423  ...cfz 12061  Diophcdioph 36202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721  ax-inf2 8295  ax-cnex 9745  ax-resscn 9746  ax-1cn 9747  ax-icn 9748  ax-addcl 9749  ax-addrcl 9750  ax-mulcl 9751  ax-mulrcl 9752  ax-mulcom 9753  ax-addass 9754  ax-mulass 9755  ax-distr 9756  ax-i2m1 9757  ax-1ne0 9758  ax-1rid 9759  ax-rnegex 9760  ax-rrecex 9761  ax-cnre 9762  ax-pre-lttri 9763  ax-pre-lttrn 9764  ax-pre-ltadd 9765  ax-pre-mulgt0 9766
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6387  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-of 6669  df-om 6832  df-1st 6932  df-2nd 6933  df-wrecs 7167  df-recs 7229  df-rdg 7267  df-1o 7321  df-oadd 7325  df-er 7503  df-map 7620  df-en 7716  df-dom 7717  df-sdom 7718  df-fin 7719  df-card 8522  df-cda 8747  df-pnf 9829  df-mnf 9830  df-xr 9831  df-ltxr 9832  df-le 9833  df-sub 10017  df-neg 10018  df-nn 10774  df-n0 11046  df-z 11117  df-uz 11424  df-fz 12062  df-hash 12844  df-mzpcl 36170  df-mzp 36171  df-dioph 36203
This theorem is referenced by:  rexfrabdioph  36243  elnn0rabdioph  36251  dvdsrabdioph  36258
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