rexrabdioph - Mathbox for Stefan O'Rear

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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 = (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 3099	. . . . . 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 3654	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
3	2	cbvrexv 3368	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ₀ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
4	3	anbi2i 616	. . . . . . . . 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 3278	. . . . . . . . 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 270	. . . . . . . 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 757	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑁 ∈ ℕ₀)
8		simpr 479	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)))
9		simplr 759	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑐 ∈ ℕ₀)
10		rexrabdioph.1	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑁 + 1)
11	10	mapfzcons 38243	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_𝑚 (1...𝑀)))
12	7, 8, 9, 11	syl3anc 1439	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_𝑚 (1...𝑀)))
13	12	adantrr 707	. . . . . . . . . . 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 38246	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
15	8, 9, 14	syl2anc 579	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
16	15	eqcomd 2784	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
17	10	mapfzcons1 38244	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
18	17	adantl 475	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
19	18	eqcomd 2784	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
20	19	sbceq1d 3657	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → ([𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
21	16, 20	sbceqbid 3659	. . . . . . . . . . . . 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 221	. . . . . . . . . . . 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 448	. . . . . . . . . . 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 707	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25		fveq1 6445	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏‘𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26		reseq1 5636	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
27	26	sbceq1d 3657	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
28	25, 27	sbceqbid 3659	. . . . . . . . . . . . 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 2788	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
30	28, 29	anbi12d 624	. . . . . . . . . . . 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 3511	. . . . . . . . . . 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 827	. . . . . . . . . 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 3216	. . . . . . . . 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 8162	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ₀)
35		nn0p1nn 11683	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
36	10, 35	syl5eqel 2863	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ ℕ)
37		elfz1end 12688	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
38	36, 37	sylib 210	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (1...𝑀))
39		ffvelrn 6621	. . . . . . . . . . . . 13 ⊢ ((𝑏:(1...𝑀)⟶ℕ₀ ∧ 𝑀 ∈ (1...𝑀)) → (𝑏‘𝑀) ∈ ℕ₀)
40	34, 38, 39	syl2anr 590	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀))) → (𝑏‘𝑀) ∈ ℕ₀)
41	40	adantr 474	. . . . . . . . . . 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 763	. . . . . . . . . . . 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 38245	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ₀ ↑_𝑚 (1...𝑁)))
44	43	ad2antlr 717	. . . . . . . . . . . 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 2859	. . . . . . . . . . 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 761	. . . . . . . . . . . 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 3654	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
48	47	sbcbidv 3702	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
49	48	ad2antll 719	. . . . . . . . . . . 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 249	. . . . . . . . . . 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 3654	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏‘𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
52	51	anbi2d 622	. . . . . . . . . . . 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 3511	. . . . . . . . . . 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 827	. . . . . . . . . 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 3216	. . . . . . . . 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 204	. . . . . . . 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 275	. . . . . . 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 2906	. . . . . 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 2826	. . . . 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 2934	. . . . . 6 ⊢ Ⅎ𝑢(ℕ₀ ↑_𝑚 (1...𝑁))
61		nfcv 2934	. . . . . 6 ⊢ Ⅎ𝑎(ℕ₀ ↑_𝑚 (1...𝑁))
62		nfv 1957	. . . . . 6 ⊢ Ⅎ𝑎∃𝑣 ∈ ℕ₀ 𝜓
63		nfcv 2934	. . . . . . 7 ⊢ Ⅎ𝑢ℕ₀
64		nfcv 2934	. . . . . . . 8 ⊢ Ⅎ𝑢𝑏
65		nfsbc1v 3672	. . . . . . . 8 ⊢ Ⅎ𝑢[𝑎 / 𝑢]𝜓
66	64, 65	nfsbc 3674	. . . . . . 7 ⊢ Ⅎ𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
67	63, 66	nfrex 3188	. . . . . 6 ⊢ Ⅎ𝑢∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
68		sbceq1a 3663	. . . . . . . 8 ⊢ (𝑢 = 𝑎 → (𝜓 ↔ [𝑎 / 𝑢]𝜓))
69	68	rexbidv 3237	. . . . . . 7 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓))
70		nfv 1957	. . . . . . . 8 ⊢ Ⅎ𝑏[𝑎 / 𝑢]𝜓
71		nfsbc1v 3672	. . . . . . . 8 ⊢ Ⅎ𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
72		sbceq1a 3663	. . . . . . . 8 ⊢ (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓 ↔ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
73	70, 71, 72	cbvrex 3364	. . . . . . 7 ⊢ (∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
74	69, 73	syl6bb 279	. . . . . 6 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
75	60, 61, 62, 67, 74	cbvrab 3395	. . . . 5 ⊢ {𝑢 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
76		fveq1 6445	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡‘𝑀) = (𝑏‘𝑀))
77		reseq1 5636	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
78	77	sbceq1d 3657	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
79	76, 78	sbceqbid 3659	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
80	79	rexrab 3580	. . . . . 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 2908	. . . . 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 2839	. . . 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 6459	. . . . . . . 8 ⊢ (𝑡‘𝑀) ∈ V
84		vex 3401	. . . . . . . . 9 ⊢ 𝑡 ∈ V
85	84	resex 5693	. . . . . . . 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 505	. . . . . . . 8 ⊢ ((𝑣 = (𝑡‘𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓 ↔ 𝜑))
89	83, 85, 88	sbc2ie 3723	. . . . . . 7 ⊢ ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑)
90	89	rabbii 3382	. . . . . 6 ⊢ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑}
91	90	rexeqi 3339	. . . . 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 2908	. . . 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	syl6eq 2830	. . 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 474	. 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 476	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ₀)
96		nn0z 11752	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
97		uzid 12007	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
98		peano2uz 12047	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑁) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
99	96, 97, 98	3syl 18	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
100	10, 99	syl5eqel 2863	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (ℤ_≥‘𝑁))
101	100	adantr 474	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑁))
102		simpr 479	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
103		diophrex 38303	. . 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 1439	. 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 2859	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 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 ℕcn 11374 ℕ₀cn0 11642 ℤcz 11728 ℤ_≥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

W3C validator