Mathbox for Stefan O'Rear < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexrabdioph Structured version   Visualization version   GIF version

Theorem rexrabdioph 36376
 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 2905 . . . . . 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 3404 . . . . . . . . . . 11 (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓[𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
32cbvrexv 3148 . . . . . . . . . 10 (∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
43anbi2i 726 . . . . . . . . 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 3073 . . . . . . . . 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 266 . . . . . . . 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 786 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑁 ∈ ℕ0)
8 simpr 476 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
9 simplr 788 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 ∈ ℕ0)
10 rexrabdioph.1 . . . . . . . . . . . . . . 15 𝑀 = (𝑁 + 1)
1110mapfzcons 36297 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℕ0𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
127, 8, 9, 11syl3anc 1318 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1312adantrr 749 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0𝑚 (1...𝑀)))
1410mapfzcons2 36300 . . . . . . . . . . . . . . . . 17 ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
158, 9, 14syl2anc 691 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
1615eqcomd 2616 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
1710mapfzcons1 36298 . . . . . . . . . . . . . . . . . 18 (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1817adantl 481 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
1918eqcomd 2616 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2019sbceq1d 3407 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0𝑚 (1...𝑁))) → ([𝑎 / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2116, 20sbceqbid 3409 . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . 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 647 . . . . . . . . . . . 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 749 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26 reseq1 5311 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
2726sbceq1d 3407 . . . . . . . . . . . . . . 15 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓[((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
2825, 27sbceqbid 3409 . . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . . 14 (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
3028, 29anbi12d 743 . . . . . . . . . . . . 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 3282 . . . . . . . . . . . 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 1316 . . . . . . . . . . 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 449 . . . . . . . . . 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 3013 . . . . . . . . 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 7765 . . . . . . . . . . . . . 14 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ0)
36 nn0p1nn 11209 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
3710, 36syl5eqel 2692 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℕ0𝑀 ∈ ℕ)
38 elfz1end 12242 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
3937, 38sylib 207 . . . . . . . . . . . . . 14 (𝑁 ∈ ℕ0𝑀 ∈ (1...𝑀))
40 ffvelrn 6265 . . . . . . . . . . . . . 14 ((𝑏:(1...𝑀)⟶ℕ0𝑀 ∈ (1...𝑀)) → (𝑏𝑀) ∈ ℕ0)
4135, 39, 40syl2anr 494 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) → (𝑏𝑀) ∈ ℕ0)
4241adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏𝑀) ∈ ℕ0)
43 simprr 792 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
4410mapfzcons1cl 36299 . . . . . . . . . . . . . 14 (𝑏 ∈ (ℕ0𝑚 (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4544ad2antlr 759 . . . . . . . . . . . . 13 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0𝑚 (1...𝑁)))
4643, 45eqeltrd 2688 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ0𝑚 (1...𝑁)))
47 simprl 790 . . . . . . . . . . . . 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 3404 . . . . . . . . . . . . . . 15 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
4948sbcbidv 3457 . . . . . . . . . . . . . 14 (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
5049ad2antll 761 . . . . . . . . . . . . 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 246 . . . . . . . . . . . 12 (((𝑁 ∈ ℕ0𝑏 ∈ (ℕ0𝑚 (1...𝑀))) ∧ ([(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)
52 dfsbcq 3404 . . . . . . . . . . . . . 14 (𝑐 = (𝑏𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓[(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
5352anbi2d 736 . . . . . . . . . . . . 13 (𝑐 = (𝑏𝑀) → ((𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∧ [(𝑏𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
5453rspcev 3282 . . . . . . . . . . . 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 1316 . . . . . . . . . . 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 449 . . . . . . . . . 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 3013 . . . . . . . . 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 201 . . . . . . . 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 271 . . . . . . 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 2728 . . . . . 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 2656 . . . . 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 2751 . . . . . 6 𝑢(ℕ0𝑚 (1...𝑁))
63 nfcv 2751 . . . . . 6 𝑎(ℕ0𝑚 (1...𝑁))
64 nfv 1830 . . . . . 6 𝑎𝑣 ∈ ℕ0 𝜓
65 nfcv 2751 . . . . . . 7 𝑢0
66 nfcv 2751 . . . . . . . 8 𝑢𝑏
67 nfsbc1v 3422 . . . . . . . 8 𝑢[𝑎 / 𝑢]𝜓
6866, 67nfsbc 3424 . . . . . . 7 𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
6965, 68nfrex 2990 . . . . . 6 𝑢𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
70 sbceq1a 3413 . . . . . . . 8 (𝑢 = 𝑎 → (𝜓[𝑎 / 𝑢]𝜓))
7170rexbidv 3034 . . . . . . 7 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓))
72 nfv 1830 . . . . . . . 8 𝑏[𝑎 / 𝑢]𝜓
73 nfsbc1v 3422 . . . . . . . 8 𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
74 sbceq1a 3413 . . . . . . . 8 (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓[𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7572, 73, 74cbvrex 3144 . . . . . . 7 (∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
7671, 75syl6bb 275 . . . . . 6 (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
7762, 63, 64, 69, 76cbvrab 3171 . . . . 5 {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
78 fveq1 6102 . . . . . . . 8 (𝑡 = 𝑏 → (𝑡𝑀) = (𝑏𝑀))
79 reseq1 5311 . . . . . . . . 9 (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
8079sbceq1d 3407 . . . . . . . 8 (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8178, 80sbceqbid 3409 . . . . . . 7 (𝑡 = 𝑏 → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓[(𝑏𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
8281rexrab 3337 . . . . . 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 2726 . . . . 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 2669 . . . 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 6113 . . . . . . . . 9 (𝑡𝑀) ∈ V
86 vex 3176 . . . . . . . . . 10 𝑡 ∈ V
8786resex 5363 . . . . . . . . 9 (𝑡 ↾ (1...𝑁)) ∈ V
88 rexrabdioph.2 . . . . . . . . . 10 (𝑣 = (𝑡𝑀) → (𝜓𝜒))
89 rexrabdioph.3 . . . . . . . . . 10 (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒𝜑))
9088, 89sylan9bb 732 . . . . . . . . 9 ((𝑣 = (𝑡𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓𝜑))
9185, 87, 90sbc2ie 3472 . . . . . . . 8 ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑)
9291a1i 11 . . . . . . 7 (𝑡 ∈ (ℕ0𝑚 (1...𝑀)) → ([(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓𝜑))
9392rabbiia 3161 . . . . . 6 {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ [(𝑡𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}
9493rexeqi 3120 . . . . 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 2726 . . . 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 2660 . . 3 (𝑁 ∈ ℕ0 → {𝑢 ∈ (ℕ0𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
9796adantr 480 . 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 472 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ0)
99 nn0z 11277 . . . . . 6 (𝑁 ∈ ℕ0𝑁 ∈ ℤ)
100 uzid 11578 . . . . . 6 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
101 peano2uz 11617 . . . . . 6 (𝑁 ∈ (ℤ𝑁) → (𝑁 + 1) ∈ (ℤ𝑁))
10299, 100, 1013syl 18 . . . . 5 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ (ℤ𝑁))
10310, 102syl5eqel 2692 . . . 4 (𝑁 ∈ ℕ0𝑀 ∈ (ℤ𝑁))
104103adantr 480 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ𝑁))
105 simpr 476 . . 3 ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
106 diophrex 36357 . . 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 1318 . 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 2688 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 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {crab 2900  [wsbc 3402   ∪ cun 3538  {csn 4125  ⟨cop 4131   ↾ cres 5040  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  1c1 9816   + caddc 9818  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  Diophcdioph 36336 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-mzpcl 36304  df-mzp 36305  df-dioph 36337 This theorem is referenced by:  rexfrabdioph  36377  elnn0rabdioph  36385  dvdsrabdioph  36392
 Copyright terms: Public domain W3C validator