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Theorem rexrabdioph 37884
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 3070 . . . . . 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 3589 . . . . . . . . . . 11 (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓[𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
32cbvrexv 3321 . . . . . . . . . 10 (∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
43anbi2i 609 . . . . . . . . 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 3240 . . . . . . . . 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 267 . . . . . . . 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 750 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑁 ∈ ℕ0)
8 simpr 471 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
9 simplr 752 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 ∈ ℕ0)
10 rexrabdioph.1 . . . . . . . . . . . . . . 15 𝑀 = (𝑁 + 1)
1110mapfzcons 37805 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
127, 8, 9, 11syl3anc 1476 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1312adantrr 696 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1410mapfzcons2 37808 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
158, 9, 14syl2anc 573 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
1615eqcomd 2777 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
1710mapfzcons1 37806 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1817adantl 467 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1918eqcomd 2777 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2019sbceq1d 3592 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ([𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2116, 20sbceqbid 3594 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . 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 442 . . . . . . . . . . . 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 696 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25 fveq1 6332 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26 reseq1 5527 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2726sbceq1d 3592 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2825, 27sbceqbid 3594 . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
3028, 29anbi12d 616 . . . . . . . . . . . . 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 3460 . . . . . . . . . . . 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 1474 . . . . . . . . . . 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 397 . . . . . . . . . 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 3179 . . . . . . . . 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 8035 . . . . . . . . . . . . . 14 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ0)
36 nn0p1nn 11539 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3710, 36syl5eqel 2854 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
38 elfz1end 12578 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
3937, 38sylib 208 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
40 ffvelrn 6502 . . . . . . . . . . . . . 14 ((𝑏:(1...𝑀)⟶ℕ0𝑀 ∈ (1...𝑀)) → (𝑏𝑀) ∈ ℕ0)
4135, 39, 40syl2anr 584 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) → (𝑏𝑀) ∈ ℕ0)
4241adantr 466 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏𝑀) ∈ ℕ0)
43 simprr 756 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
4410mapfzcons1cl 37807 . . . . . . . . . . . . . 14 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4544ad2antlr 706 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4643, 45eqeltrd 2850 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
47 simprl 754 . . . . . . . . . . . . 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 3589 . . . . . . . . . . . . . . 15 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4948sbcbidv 3642 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
5049ad2antll 708 . . . . . . . . . . . . 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 247 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)
52 dfsbcq 3589 . . . . . . . . . . . . . 14 (𝑐 = (𝑏𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
5352anbi2d 614 . . . . . . . . . . . . 13 (𝑐 = (𝑏𝑀) → ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
5453rspcev 3460 . . . . . . . . . . . 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 1474 . . . . . . . . . . 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 397 . . . . . . . . . 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 3179 . . . . . . . . 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 202 . . . . . . . 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 272 . . . . . . 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 2890 . . . . . 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 2817 . . . . 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 2913 . . . . . 6 𝑢(ℕ0𝑚 (1...𝑁))
63 nfcv 2913 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
64 nfv 1995 . . . . . 6 𝑎𝑣 ∈ ℕ0 𝜓
65 nfcv 2913 . . . . . . 7 𝑢0
66 nfcv 2913 . . . . . . . 8 𝑢𝑏
67 nfsbc1v 3607 . . . . . . . 8 𝑢[𝑎 / 𝑢]𝜓
6866, 67nfsbc 3609 . . . . . . 7 𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
6965, 68nfrex 3155 . . . . . 6 𝑢𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
70 sbceq1a 3598 . . . . . . . 8 (𝑢 = 𝑎 → (𝜓[𝑎 / 𝑢]𝜓))
7170rexbidv 3200 . . . . . . 7 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓))
72 nfv 1995 . . . . . . . 8 𝑏[𝑎 / 𝑢]𝜓
73 nfsbc1v 3607 . . . . . . . 8 𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
74 sbceq1a 3598 . . . . . . . 8 (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓[𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7572, 73, 74cbvrex 3317 . . . . . . 7 (∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
7671, 75syl6bb 276 . . . . . 6 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7762, 63, 64, 69, 76cbvrab 3348 . . . . 5 {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
78 fveq1 6332 . . . . . . . 8 (𝑡 = 𝑏 → (𝑡𝑀) = (𝑏𝑀))
79 reseq1 5527 . . . . . . . . 9 (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
8079sbceq1d 3592 . . . . . . . 8 (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8178, 80sbceqbid 3594 . . . . . . 7 (𝑡 = 𝑏 → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8281rexrab 3522 . . . . . 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 2888 . . . . 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 2830 . . . 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 6344 . . . . . . . 8 (𝑡𝑀) ∈ V
86 vex 3354 . . . . . . . . 9 𝑡 ∈ V
8786resex 5583 . . . . . . . 8 (𝑡 ↾ (1...𝑁)) ∈ V
88 rexrabdioph.2 . . . . . . . . 9 (𝑣 = (𝑡𝑀) → (𝜓𝜒))
89 rexrabdioph.3 . . . . . . . . 9 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))
9088, 89sylan9bb 499 . . . . . . . 8 ((𝑣 = (𝑡𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓𝜑))
9185, 87, 90sbc2ie 3655 . . . . . . 7 ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑)
9291rabbii 3335 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}
9392rexeqi 3292 . . . . 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)))
9493abbii 2888 . . . 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))}
9584, 94syl6eq 2821 . . 3 (𝑁 ∈ ℕ0 → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
9695adantr 466 . 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))})
97 simpl 468 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ0)
98 nn0z 11607 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
99 uzid 11908 . . . . . 6 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
100 peano2uz 11948 . . . . . 6 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
10198, 99, 1003syl 18 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (ℤ𝑁))
10210, 101syl5eqel 2854 . . . 4 (𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁))
103102adantr 466 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ𝑁))
104 simpr 471 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
105 diophrex 37865 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
10697, 103, 104, 105syl3anc 1476 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
10796, 106eqeltrd 2850 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 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  {crab 3065  [wsbc 3587  cun 3721  {csn 4317  cop 4323  cres 5252  wf 6026  cfv 6030  (class class class)co 6796  𝑚 cmap 8013  1c1 10143   + caddc 10145  cn 11226  0cn0 11499  cz 11584  cuz 11893  ...cfz 12533  Diophcdioph 37844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-inf2 8706  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-of 7048  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-er 7900  df-map 8015  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-card 8969  df-cda 9196  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-n0 11500  df-z 11585  df-uz 11894  df-fz 12534  df-hash 13322  df-mzpcl 37812  df-mzp 37813  df-dioph 37845
This theorem is referenced by:  rexfrabdioph  37885  elnn0rabdioph  37893  dvdsrabdioph  37900
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