rexrabdioph - Mathbox for Stefan O'Rear

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Theorem rexrabdioph 40208

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 3062	. . . . . 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 3682	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
3	2	cbvrexvw 3350	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ₀ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
4	3	anbi2i 626	. . . . . . . . 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 3254	. . . . . . . . 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 767	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑐 ∈ ℕ₀)
10		rexrabdioph.1	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑁 + 1)
11	10	mapfzcons 40130	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_m (1...𝑀)))
12	7, 8, 9, 11	syl3anc 1372	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_m (1...𝑀)))
13	12	adantrr 717	. . . . . . . . . . 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 40133	. . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
17	10	mapfzcons1 40131	. . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
20	19	sbceq1d 3685	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_m (1...𝑁))) → ([𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
21	16, 20	sbceqbid 3687	. . . . . . . . . . . . 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 717	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25		fveq1 6673	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏‘𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26		reseq1 5819	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
27	26	sbceq1d 3685	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
28	25, 27	sbceqbid 3687	. . . . . . . . . . . . 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 2749	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
30	28, 29	anbi12d 634	. . . . . . . . . . . 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 3526	. . . . . . . . . . 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 836	. . . . . . . . . 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 3197	. . . . . . . . 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 8459	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ₀)
35		nn0p1nn 12015	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
36	10, 35	eqeltrid 2837	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ ℕ)
37		elfz1end 13028	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
38	36, 37	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (1...𝑀))
39		ffvelrn 6859	. . . . . . . . . . . . 13 ⊢ ((𝑏:(1...𝑀)⟶ℕ₀ ∧ 𝑀 ∈ (1...𝑀)) → (𝑏‘𝑀) ∈ ℕ₀)
40	34, 38, 39	syl2anr 600	. . . . . . . . . . . 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 773	. . . . . . . . . . . 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 40132	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ₀ ↑_m (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ₀ ↑_m (1...𝑁)))
44	43	ad2antlr 727	. . . . . . . . . . . 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 2833	. . . . . . . . . . 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 771	. . . . . . . . . . . 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 3682	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
48	47	sbcbidv 3736	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
49	48	ad2antll 729	. . . . . . . . . . . 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 3682	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏‘𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
52	51	anbi2d 632	. . . . . . . . . . . 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 3526	. . . . . . . . . . 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 836	. . . . . . . . . 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 3197	. . . . . . . . 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 2802	. . . . . 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 2785	. . . . 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 2899	. . . . . 6 ⊢ Ⅎ𝑢(ℕ₀ ↑_m (1...𝑁))
61		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑎(ℕ₀ ↑_m (1...𝑁))
62		nfv 1921	. . . . . 6 ⊢ Ⅎ𝑎∃𝑣 ∈ ℕ₀ 𝜓
63		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑢ℕ₀
64		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑢𝑏
65		nfsbc1v 3700	. . . . . . . 8 ⊢ Ⅎ𝑢[𝑎 / 𝑢]𝜓
66	64, 65	nfsbcw 3702	. . . . . . 7 ⊢ Ⅎ𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
67	63, 66	nfrex 3219	. . . . . 6 ⊢ Ⅎ𝑢∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
68		sbceq1a 3691	. . . . . . . 8 ⊢ (𝑢 = 𝑎 → (𝜓 ↔ [𝑎 / 𝑢]𝜓))
69	68	rexbidv 3207	. . . . . . 7 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓))
70		nfv 1921	. . . . . . . 8 ⊢ Ⅎ𝑏[𝑎 / 𝑢]𝜓
71		nfsbc1v 3700	. . . . . . . 8 ⊢ Ⅎ𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
72		sbceq1a 3691	. . . . . . . 8 ⊢ (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓 ↔ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
73	70, 71, 72	cbvrexw 3341	. . . . . . 7 ⊢ (∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
74	69, 73	bitrdi 290	. . . . . 6 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
75	60, 61, 62, 67, 74	cbvrabw 3391	. . . . 5 ⊢ {𝑢 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∈ (ℕ₀ ↑_m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
76		fveq1 6673	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡‘𝑀) = (𝑏‘𝑀))
77		reseq1 5819	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
78	77	sbceq1d 3685	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
79	76, 78	sbceqbid 3687	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
80	79	rexrab 3595	. . . . . 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 2803	. . . . 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 2798	. . . 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 6687	. . . . . . . 8 ⊢ (𝑡‘𝑀) ∈ V
84		vex 3402	. . . . . . . . 9 ⊢ 𝑡 ∈ V
85	84	resex 5873	. . . . . . . 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 3758	. . . . . . 7 ⊢ ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑)
90	89	rabbii 3374	. . . . . 6 ⊢ {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ₀ ↑_m (1...𝑀)) ∣ 𝜑}
91	90	rexeqi 3315	. . . . 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 2803	. . . 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 2789	. . 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 12086	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
97		uzid 12339	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
98		peano2uz 12383	. . . . . 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 2837	. . . 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 40189	. . 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 1372	. 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 2833	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 1542 ∈ wcel 2114 {cab 2716 ∃wrex 3054 {crab 3057 [wsbc 3680 ∪ cun 3841 {csn 4516 ⟨cop 4522 ↾ cres 5527 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ↑_m cmap 8437 1c1 10616 + caddc 10618 ℕcn 11716 ℕ₀cn0 11976 ℤcz 12062 ℤ_≥cuz 12324 ...cfz 12981 Diophcdioph 40169

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-inf2 9177 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-oadd 8135 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-dju 9403 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-hash 13783 df-mzpcl 40137 df-mzp 40138 df-dioph 40170

This theorem is referenced by: rexfrabdioph 40209 elnn0rabdioph 40217 dvdsrabdioph 40224

W3C validator