rexrabdioph - Mathbox for Stefan O'Rear

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Theorem rexrabdioph 39735

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 = (𝑡‘𝑀) → (𝜓 ↔ 𝜒))
rexrabdioph.3	⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒 ↔ 𝜑))

**Assertion**
Ref	Expression
rexrabdioph	⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ₀ ↑_m (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.

Step	Hyp	Ref	Expression
1		df-rab 3115	. . . . . 6 ⊢ {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓} = {𝑎 ∣ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)}
2		dfsbcq 3722	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
3	2	cbvrexvw 3397	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ₀ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
4	3	anbi2i 625	. . . . . . . . 9 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑐 ∈ ℕ₀ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
5		r19.42v 3303	. . . . . . . . 9 ⊢ (∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑐 ∈ ℕ₀ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
6	4, 5	bitr4i 281	. . . . . . . 8 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
7		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑁 ∈ ℕ₀)
8		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)))
9		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑐 ∈ ℕ₀)
10		rexrabdioph.1	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑁 + 1)
11	10	mapfzcons 39657	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_m (1...𝑀)))
12	7, 8, 9, 11	syl3anc 1368	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_m (1...𝑀)))
13	12	adantrr 716	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_m (1...𝑀)))
14	10	mapfzcons2 39660	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
15	8, 9, 14	syl2anc 587	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
16	15	eqcomd 2804	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
17	10	mapfzcons1 39658	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
18	17	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
19	18	eqcomd 2804	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
20	19	sbceq1d 3725	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → ([𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
21	16, 20	sbceqbid 3727	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
22	21	biimpd 232	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 → [((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
23	22	impr 458	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → [((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓)
24	19	adantrr 716	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25		fveq1 6644	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏‘𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26		reseq1 5812	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
27	26	sbceq1d 3725	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
28	25, 27	sbceqbid 3727	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
29	26	eqeq2d 2809	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
30	28, 29	anbi12d 633	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) ↔ ([((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))))
31	30	rspcev 3571	. . . . . . . . . . 11 ⊢ (((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_m (1...𝑀)) ∧ ([((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) / 𝑣][((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))) → ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
32	13, 23, 24, 31	syl12anc 835	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
33	32	rexlimdva2 3246	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) → ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
34		elmapi 8411	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ₀)
35		nn0p1nn 11924	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
36	10, 35	eqeltrid 2894	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ ℕ)
37		elfz1end 12932	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
38	36, 37	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (1...𝑀))
39		ffvelrn 6826	. . . . . . . . . . . . 13 ⊢ ((𝑏:(1...𝑀)⟶ℕ₀ ∧ 𝑀 ∈ (1...𝑀)) → (𝑏‘𝑀) ∈ ℕ₀)
40	34, 38, 39	syl2anr 599	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) → (𝑏‘𝑀) ∈ ℕ₀)
41	40	adantr 484	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏‘𝑀) ∈ ℕ₀)
42		simprr 772	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
43	10	mapfzcons1cl 39659	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ₀ ↑_m (1...𝑁)))
44	43	ad2antlr 726	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ₀ ↑_m (1...𝑁)))
45	42, 44	eqeltrd 2890	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)))
46		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓)
47		dfsbcq 3722	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
48	47	sbcbidv 3774	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
49	48	ad2antll 728	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
50	46, 49	mpbird 260	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓)
51		dfsbcq 3722	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏‘𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
52	51	anbi2d 631	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑏‘𝑀) → ((𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
53	52	rspcev 3571	. . . . . . . . . . 11 ⊢ (((𝑏‘𝑀) ∈ ℕ₀ ∧ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓)) → ∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
54	41, 45, 50, 53	syl12anc 835	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → ∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
55	54	rexlimdva2 3246	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → (∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))) → ∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)))
56	33, 55	impbid 215	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ₀ → (∃𝑐 ∈ ℕ₀ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
57	6, 56	syl5bb 286	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → ((𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓) ↔ ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))))
58	57	abbidv 2862	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → {𝑎 ∣ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)} = {𝑎 ∣ ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))})
59	1, 58	syl5eq 2845	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓} = {𝑎 ∣ ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))})
60		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑢(ℕ₀ ↑_m (1...𝑁))
61		nfcv 2955	. . . . . 6 ⊢ Ⅎ𝑎(ℕ₀ ↑_m (1...𝑁))
62		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑎∃𝑣 ∈ ℕ₀ 𝜓
63		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑢ℕ₀
64		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑢𝑏
65		nfsbc1v 3740	. . . . . . . 8 ⊢ Ⅎ𝑢[𝑎 / 𝑢]𝜓
66	64, 65	nfsbcw 3742	. . . . . . 7 ⊢ Ⅎ𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
67	63, 66	nfrex 3268	. . . . . 6 ⊢ Ⅎ𝑢∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
68		sbceq1a 3731	. . . . . . . 8 ⊢ (𝑢 = 𝑎 → (𝜓 ↔ [𝑎 / 𝑢]𝜓))
69	68	rexbidv 3256	. . . . . . 7 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓))
70		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑏[𝑎 / 𝑢]𝜓
71		nfsbc1v 3740	. . . . . . . 8 ⊢ Ⅎ𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
72		sbceq1a 3731	. . . . . . . 8 ⊢ (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓 ↔ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
73	70, 71, 72	cbvrexw 3388	. . . . . . 7 ⊢ (∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
74	69, 73	syl6bb 290	. . . . . 6 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
75	60, 61, 62, 67, 74	cbvrabw 3437	. . . . 5 ⊢ {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
76		fveq1 6644	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡‘𝑀) = (𝑏‘𝑀))
77		reseq1 5812	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
78	77	sbceq1d 3725	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
79	76, 78	sbceqbid 3727	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
80	79	rexrab 3635	. . . . . 6 ⊢ (∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁))))
81	80	abbii 2863	. . . . 5 ⊢ {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ (ℕ₀ ↑_m (1...𝑀))([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))}
82	59, 75, 81	3eqtr4g 2858	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁))})
83		fvex 6658	. . . . . . . 8 ⊢ (𝑡‘𝑀) ∈ V
84		vex 3444	. . . . . . . . 9 ⊢ 𝑡 ∈ V
85	84	resex 5866	. . . . . . . 8 ⊢ (𝑡 ↾ (1...𝑁)) ∈ V
86		rexrabdioph.2	. . . . . . . . 9 ⊢ (𝑣 = (𝑡‘𝑀) → (𝜓 ↔ 𝜒))
87		rexrabdioph.3	. . . . . . . . 9 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒 ↔ 𝜑))
88	86, 87	sylan9bb 513	. . . . . . . 8 ⊢ ((𝑣 = (𝑡‘𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓 ↔ 𝜑))
89	83, 85, 88	sbc2ie 3796	. . . . . . 7 ⊢ ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑)
90	89	rabbii 3420	. . . . . 6 ⊢ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}
91	90	rexeqi 3363	. . . . 5 ⊢ (∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁)) ↔ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁)))
92	91	abbii 2863	. . . 4 ⊢ {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓}𝑎 = (𝑏 ↾ (1...𝑁))} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))}
93	82, 92	eqtrdi 2849	. . 3 ⊢ (𝑁 ∈ ℕ₀ → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
94	93	adantr 484	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
95		simpl 486	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ₀)
96		nn0z 11993	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
97		uzid 12246	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
98		peano2uz 12289	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑁) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
99	96, 97, 98	3syl 18	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
100	10, 99	eqeltrid 2894	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (ℤ_≥‘𝑁))
101	100	adantr 484	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑁))
102		simpr 488	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
103		diophrex 39716	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑀 ∈ (ℤ_≥‘𝑁) ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
104	95, 101, 102, 103	syl3anc 1368	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))} ∈ (Dioph‘𝑁))
105	94, 104	eqeltrd 2890	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} ∈ (Dioph‘𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2776 ∃wrex 3107 {crab 3110 [wsbc 3720 ∪ cun 3879 {csn 4525 ⟨cop 4531 ↾ cres 5521 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_m cmap 8389 1c1 10527 + caddc 10529 ℕcn 11625 ℕ₀cn0 11885 ℤcz 11969 ℤ_≥cuz 12231 ...cfz 12885 Diophcdioph 39696

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 df-mzpcl 39664 df-mzp 39665 df-dioph 39697

This theorem is referenced by: rexfrabdioph 39736 elnn0rabdioph 39744 dvdsrabdioph 39751

W3C validator