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Theorem rexrabdioph 39391
 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 3147 . . . . . 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 3773 . . . . . . . . . . 11 (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓[𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
32cbvrexvw 3450 . . . . . . . . . 10 (∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
43anbi2i 624 . . . . . . . . 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 3350 . . . . . . . . 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 280 . . . . . . . 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 765 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → 𝑁 ∈ ℕ0)
8 simpr 487 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → 𝑎 ∈ (ℕ0m (1...𝑁)))
9 simplr 767 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → 𝑐 ∈ ℕ0)
10 rexrabdioph.1 . . . . . . . . . . . . . 14 𝑀 = (𝑁 + 1)
1110mapfzcons 39313 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑎 ∈ (ℕ0m (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0m (1...𝑀)))
127, 8, 9, 11syl3anc 1367 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0m (1...𝑀)))
1312adantrr 715 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0m (1...𝑀)))
1410mapfzcons2 39316 . . . . . . . . . . . . . . . 16 ((𝑎 ∈ (ℕ0m (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
158, 9, 14syl2anc 586 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
1615eqcomd 2827 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
1710mapfzcons1 39314 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ (ℕ0m (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1817adantl 484 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1918eqcomd 2827 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2019sbceq1d 3776 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0m (1...𝑁))) → ([𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2116, 20sbceqbid 3778 . . . . . . . . . . . . 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 231 . . . . . . . . . . . 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 457 . . . . . . . . . . 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 715 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25 fveq1 6668 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26 reseq1 5846 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2726sbceq1d 3776 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2825, 27sbceqbid 3778 . . . . . . . . . . . . 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 2832 . . . . . . . . . . . . 13 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
3028, 29anbi12d 632 . . . . . . . . . . . 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 3622 . . . . . . . . . . 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 834 . . . . . . . . . 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 3287 . . . . . . . . 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 8427 . . . . . . . . . . . . 13 (𝑏 ∈ (ℕ0m (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ0)
35 nn0p1nn 11935 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3610, 35eqeltrid 2917 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
37 elfz1end 12936 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
3836, 37sylib 220 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
39 ffvelrn 6848 . . . . . . . . . . . . 13 ((𝑏:(1...𝑀)⟶ℕ0𝑀 ∈ (1...𝑀)) → (𝑏𝑀) ∈ ℕ0)
4034, 38, 39syl2anr 598 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0m (1...𝑀))) → (𝑏𝑀) ∈ ℕ0)
4140adantr 483 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0m (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏𝑀) ∈ ℕ0)
42 simprr 771 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0m (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
4310mapfzcons1cl 39315 . . . . . . . . . . . . 13 (𝑏 ∈ (ℕ0m (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)))
4443ad2antlr 725 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0m (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0m (1...𝑁)))
4542, 44eqeltrd 2913 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0m (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ0m (1...𝑁)))
46 simprl 769 . . . . . . . . . . . 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 3773 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4847sbcbidv 3826 . . . . . . . . . . . . 13 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4948ad2antll 727 . . . . . . . . . . . 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 259 . . . . . . . . . . 11 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0m (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)
51 dfsbcq 3773 . . . . . . . . . . . . 13 (𝑐 = (𝑏𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
5251anbi2d 630 . . . . . . . . . . . 12 (𝑐 = (𝑏𝑀) → ((𝑎 ∈ (ℕ0m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0m (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
5352rspcev 3622 . . . . . . . . . . 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 834 . . . . . . . . . 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 3287 . . . . . . . . 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 214 . . . . . . . 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 285 . . . . . . 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 2885 . . . . . 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 2868 . . . . 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 2977 . . . . . 6 𝑢(ℕ0m (1...𝑁))
61 nfcv 2977 . . . . . 6 𝑎(ℕ0m (1...𝑁))
62 nfv 1911 . . . . . 6 𝑎𝑣 ∈ ℕ0 𝜓
63 nfcv 2977 . . . . . . 7 𝑢0
64 nfcv 2977 . . . . . . . 8 𝑢𝑏
65 nfsbc1v 3791 . . . . . . . 8 𝑢[𝑎 / 𝑢]𝜓
6664, 65nfsbcw 3793 . . . . . . 7 𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
6763, 66nfrex 3309 . . . . . 6 𝑢𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
68 sbceq1a 3782 . . . . . . . 8 (𝑢 = 𝑎 → (𝜓[𝑎 / 𝑢]𝜓))
6968rexbidv 3297 . . . . . . 7 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓))
70 nfv 1911 . . . . . . . 8 𝑏[𝑎 / 𝑢]𝜓
71 nfsbc1v 3791 . . . . . . . 8 𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
72 sbceq1a 3782 . . . . . . . 8 (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓[𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7370, 71, 72cbvrexw 3442 . . . . . . 7 (∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
7469, 73syl6bb 289 . . . . . 6 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7560, 61, 62, 67, 74cbvrabw 3489 . . . . 5 {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
76 fveq1 6668 . . . . . . . 8 (𝑡 = 𝑏 → (𝑡𝑀) = (𝑏𝑀))
77 reseq1 5846 . . . . . . . . 9 (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
7877sbceq1d 3776 . . . . . . . 8 (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
7976, 78sbceqbid 3778 . . . . . . 7 (𝑡 = 𝑏 → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8079rexrab 3686 . . . . . 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 2886 . . . . 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 2881 . . . 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 6682 . . . . . . . 8 (𝑡𝑀) ∈ V
84 vex 3497 . . . . . . . . 9 𝑡 ∈ V
8584resex 5898 . . . . . . . 8 (𝑡 ↾ (1...𝑁)) ∈ V
86 rexrabdioph.2 . . . . . . . . 9 (𝑣 = (𝑡𝑀) → (𝜓𝜒))
87 rexrabdioph.3 . . . . . . . . 9 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))
8886, 87sylan9bb 512 . . . . . . . 8 ((𝑣 = (𝑡𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓𝜑))
8983, 85, 88sbc2ie 3849 . . . . . . 7 ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑)
9089rabbii 3473 . . . . . 6 {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑}
9190rexeqi 3414 . . . . 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 2886 . . . 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 2872 . . 3 (𝑁 ∈ ℕ0 → {𝑢 ∈ (ℕ0m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
9493adantr 483 . 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 485 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ0)
96 nn0z 12004 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
97 uzid 12257 . . . . . 6 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
98 peano2uz 12300 . . . . . 6 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
9996, 97, 983syl 18 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (ℤ𝑁))
10010, 99eqeltrid 2917 . . . 4 (𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁))
101100adantr 483 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ𝑁))
102 simpr 487 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
103 diophrex 39372 . . 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 1367 . 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 2913 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 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  ∃wrex 3139  {crab 3142  [wsbc 3771   ∪ cun 3933  {csn 4566  ⟨cop 4572   ↾ cres 5556  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155   ↑m cmap 8405  1c1 10537   + caddc 10539  ℕcn 11637  ℕ0cn0 11896  ℤcz 11980  ℤ≥cuz 12242  ...cfz 12891  Diophcdioph 39352 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-inf2 9103  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-of 7408  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-dju 9329  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-hash 13690  df-mzpcl 39320  df-mzp 39321  df-dioph 39353 This theorem is referenced by:  rexfrabdioph  39392  elnn0rabdioph  39400  dvdsrabdioph  39407
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