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Theorem rexrabdioph 38322
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 3099 . . . . . 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 3654 . . . . . . . . . . 11 (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓[𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
32cbvrexv 3368 . . . . . . . . . 10 (∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
43anbi2i 616 . . . . . . . . 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 3278 . . . . . . . . 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 270 . . . . . . . 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 757 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑁 ∈ ℕ0)
8 simpr 479 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
9 simplr 759 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 ∈ ℕ0)
10 rexrabdioph.1 . . . . . . . . . . . . . 14 𝑀 = (𝑁 + 1)
1110mapfzcons 38243 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
127, 8, 9, 11syl3anc 1439 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1312adantrr 707 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1410mapfzcons2 38246 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
158, 9, 14syl2anc 579 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
1615eqcomd 2784 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
1710mapfzcons1 38244 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1817adantl 475 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1918eqcomd 2784 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2019sbceq1d 3657 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ([𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2116, 20sbceqbid 3659 . . . . . . . . . . . . 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))
2221biimpd 221 . . . . . . . . . . . 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))
2322impr 448 . . . . . . . . . . 11 (((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 707 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25 fveq1 6445 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26 reseq1 5636 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2726sbceq1d 3657 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2825, 27sbceqbid 3659 . . . . . . . . . . . . 13 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2926eqeq2d 2788 . . . . . . . . . . . . 13 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
3028, 29anbi12d 624 . . . . . . . . . . . 12 (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 3511 . . . . . . . . . . 11 (((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 827 . . . . . . . . . 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))))
3332rexlimdva2 3216 . . . . . . . . 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)))))
34 elmapi 8162 . . . . . . . . . . . . 13 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ0)
35 nn0p1nn 11683 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3610, 35syl5eqel 2863 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
37 elfz1end 12688 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
3836, 37sylib 210 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
39 ffvelrn 6621 . . . . . . . . . . . . 13 ((𝑏:(1...𝑀)⟶ℕ0𝑀 ∈ (1...𝑀)) → (𝑏𝑀) ∈ ℕ0)
4034, 38, 39syl2anr 590 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) → (𝑏𝑀) ∈ ℕ0)
4140adantr 474 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏𝑀) ∈ ℕ0)
42 simprr 763 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
4310mapfzcons1cl 38245 . . . . . . . . . . . . 13 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4443ad2antlr 717 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4542, 44eqeltrd 2859 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
46 simprl 761 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)
47 dfsbcq 3654 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4847sbcbidv 3702 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4948ad2antll 719 . . . . . . . . . . . 12 (((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))
5046, 49mpbird 249 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)
51 dfsbcq 3654 . . . . . . . . . . . . 13 (𝑐 = (𝑏𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
5251anbi2d 622 . . . . . . . . . . . 12 (𝑐 = (𝑏𝑀) → ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
5352rspcev 3511 . . . . . . . . . . 11 (((𝑏𝑀) ∈ ℕ0 ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)) → ∃𝑐 ∈ ℕ0 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
5441, 45, 50, 53syl12anc 827 . . . . . . . . . 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))
5554rexlimdva2 3216 . . . . . . . . 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)))
5633, 55impbid 204 . . . . . . . 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)))))
576, 56syl5bb 275 . . . . . . 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)))))
5857abbidv 2906 . . . . . 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)))})
591, 58syl5eq 2826 . . . . 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)))})
60 nfcv 2934 . . . . . 6 𝑢(ℕ0𝑚 (1...𝑁))
61 nfcv 2934 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
62 nfv 1957 . . . . . 6 𝑎𝑣 ∈ ℕ0 𝜓
63 nfcv 2934 . . . . . . 7 𝑢0
64 nfcv 2934 . . . . . . . 8 𝑢𝑏
65 nfsbc1v 3672 . . . . . . . 8 𝑢[𝑎 / 𝑢]𝜓
6664, 65nfsbc 3674 . . . . . . 7 𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
6763, 66nfrex 3188 . . . . . 6 𝑢𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
68 sbceq1a 3663 . . . . . . . 8 (𝑢 = 𝑎 → (𝜓[𝑎 / 𝑢]𝜓))
6968rexbidv 3237 . . . . . . 7 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓))
70 nfv 1957 . . . . . . . 8 𝑏[𝑎 / 𝑢]𝜓
71 nfsbc1v 3672 . . . . . . . 8 𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
72 sbceq1a 3663 . . . . . . . 8 (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓[𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7370, 71, 72cbvrex 3364 . . . . . . 7 (∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
7469, 73syl6bb 279 . . . . . 6 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7560, 61, 62, 67, 74cbvrab 3395 . . . . 5 {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
76 fveq1 6445 . . . . . . . 8 (𝑡 = 𝑏 → (𝑡𝑀) = (𝑏𝑀))
77 reseq1 5636 . . . . . . . . 9 (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
7877sbceq1d 3657 . . . . . . . 8 (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
7976, 78sbceqbid 3659 . . . . . . 7 (𝑡 = 𝑏 → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8079rexrab 3580 . . . . . 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))))
8180abbii 2908 . . . . 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)))}
8259, 75, 813eqtr4g 2839 . . . 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))})
83 fvex 6459 . . . . . . . 8 (𝑡𝑀) ∈ V
84 vex 3401 . . . . . . . . 9 𝑡 ∈ V
8584resex 5693 . . . . . . . 8 (𝑡 ↾ (1...𝑁)) ∈ V
86 rexrabdioph.2 . . . . . . . . 9 (𝑣 = (𝑡𝑀) → (𝜓𝜒))
87 rexrabdioph.3 . . . . . . . . 9 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))
8886, 87sylan9bb 505 . . . . . . . 8 ((𝑣 = (𝑡𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓𝜑))
8983, 85, 88sbc2ie 3723 . . . . . . 7 ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑)
9089rabbii 3382 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}
9190rexeqi 3339 . . . . 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)))
9291abbii 2908 . . . 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))}
9382, 92syl6eq 2830 . . 3 (𝑁 ∈ ℕ0 → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
9493adantr 474 . 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))})
95 simpl 476 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ0)
96 nn0z 11752 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
97 uzid 12007 . . . . . 6 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
98 peano2uz 12047 . . . . . 6 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
9996, 97, 983syl 18 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (ℤ𝑁))
10010, 99syl5eqel 2863 . . . 4 (𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁))
101100adantr 474 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ𝑁))
102 simpr 479 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
103 diophrex 38303 . . 3 ((𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁) ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
10495, 101, 102, 103syl3anc 1439 . 2 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
10594, 104eqeltrd 2859 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 198  wa 386   = wceq 1601  wcel 2107  {cab 2763  wrex 3091  {crab 3094  [wsbc 3652  cun 3790  {csn 4398  cop 4404  cres 5357  wf 6131  cfv 6135  (class class class)co 6922  𝑚 cmap 8140  1c1 10273   + caddc 10275  cn 11374  0cn0 11642  cz 11728  cuz 11992  ...cfz 12643  Diophcdioph 38282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-cda 9325  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-hash 13436  df-mzpcl 38250  df-mzp 38251  df-dioph 38283
This theorem is referenced by:  rexfrabdioph  38323  elnn0rabdioph  38331  dvdsrabdioph  38338
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