Theorem rexrabdioph 40616

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 3073	. . . . . 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 3718	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
3	2	cbvrexvw 3384	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ0 [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
4	3	anbi2i 623	. . . . . . . . 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 3279	. . . . . . . . 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 277	. . . . . . . 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 764	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑁))) → 𝑁 ∈ ℕ0)
8		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑁))) → 𝑎 ∈ (ℕ0 ↑m (1...𝑁)))
9		simplr 766	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑁))) → 𝑐 ∈ ℕ0)
10		rexrabdioph.1	. . . . . . . . . . . . . 14 ⊢ 𝑀 = (𝑁 + 1)
11	10	mapfzcons 40538	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0 ↑m (1...𝑀)))
12	7, 8, 9, 11	syl3anc 1370	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ0 ↑m (1...𝑀)))
13	12	adantrr 714	. . . . . . . . . . 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 40541	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ 𝑐 ∈ ℕ0) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
15	8, 9, 14	syl2anc 584	. . . . . . . . . . . . . . 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 40539	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 ∈ (ℕ0 ↑m (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
18	17	adantl 482	. . . . . . . . . . . . . . . 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 3721	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) ∧ 𝑎 ∈ (ℕ0 ↑m (1...𝑁))) → ([𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
21	16, 20	sbceqbid 3723	. . . . . . . . . . . . 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 228	. . . . . . . . . . . 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 455	. . . . . . . . . . 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 714	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) ∧ (𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25		fveq1 6773	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏‘𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26		reseq1 5885	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
27	26	sbceq1d 3721	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
28	25, 27	sbceqbid 3723	. . . . . . . . . . . . 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 631	. . . . . . . . . . . 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 3561	. . . . . . . . . . 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 834	. . . . . . . . . 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 8637	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ0)
35		nn0p1nn 12272	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ)
36	10, 35	eqeltrid 2843	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ ℕ)
37		elfz1end 13286	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
38	36, 37	sylib 217	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ (1...𝑀))
39		ffvelrn 6959	. . . . . . . . . . . . 13 ⊢ ((𝑏:(1...𝑀)⟶ℕ0 ∧ 𝑀 ∈ (1...𝑀)) → (𝑏‘𝑀) ∈ ℕ0)
40	34, 38, 39	syl2anr 597	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑏 ∈ (ℕ0 ↑m (1...𝑀))) → (𝑏‘𝑀) ∈ ℕ0)
41	40	adantr 481	. . . . . . . . . . 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 770	. . . . . . . . . . . 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 40540	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (ℕ0 ↑m (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ0 ↑m (1...𝑁)))
44	43	ad2antlr 724	. . . . . . . . . . . 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 2839	. . . . . . . . . . 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 768	. . . . . . . . . . . 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 3718	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
48	47	sbcbidv 3775	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
49	48	ad2antll 726	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 3718	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏‘𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
52	51	anbi2d 629	. . . . . . . . . . . 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 3561	. . . . . . . . . . 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 834	. . . . . . . . . 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 211	. . . . . . . 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 283	. . . . . . 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 2807	. . . . . 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	eqtrid 2790	. . . . 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 2907	. . . . . 6 ⊢ Ⅎ𝑢(ℕ0 ↑m (1...𝑁))
61		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑎(ℕ0 ↑m (1...𝑁))
62		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑎∃𝑣 ∈ ℕ0 𝜓
63		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑢ℕ0
64		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑢𝑏
65		nfsbc1v 3736	. . . . . . . 8 ⊢ Ⅎ𝑢[𝑎 / 𝑢]𝜓
66	64, 65	nfsbcw 3738	. . . . . . 7 ⊢ Ⅎ𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
67	63, 66	nfrex 3242	. . . . . 6 ⊢ Ⅎ𝑢∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
68		sbceq1a 3727	. . . . . . . 8 ⊢ (𝑢 = 𝑎 → (𝜓 ↔ [𝑎 / 𝑢]𝜓))
69	68	rexbidv 3226	. . . . . . 7 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓))
70		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑏[𝑎 / 𝑢]𝜓
71		nfsbc1v 3736	. . . . . . . 8 ⊢ Ⅎ𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
72		sbceq1a 3727	. . . . . . . 8 ⊢ (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓 ↔ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
73	70, 71, 72	cbvrexw 3374	. . . . . . 7 ⊢ (∃𝑣 ∈ ℕ0 [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
74	69, 73	bitrdi 287	. . . . . 6 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ0 𝜓 ↔ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
75	60, 61, 62, 67, 74	cbvrabw 3424	. . . . 5 ⊢ {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} = {𝑎 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑏 ∈ ℕ0 [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
76		fveq1 6773	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡‘𝑀) = (𝑏‘𝑀))
77		reseq1 5885	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
78	77	sbceq1d 3721	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
79	76, 78	sbceqbid 3723	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
80	79	rexrab 3633	. . . . . 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 2808	. . . . 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 2803	. . . 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 6787	. . . . . . . 8 ⊢ (𝑡‘𝑀) ∈ V
84		vex 3436	. . . . . . . . 9 ⊢ 𝑡 ∈ V
85	84	resex 5939	. . . . . . . 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 510	. . . . . . . 8 ⊢ ((𝑣 = (𝑡‘𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓 ↔ 𝜑))
89	83, 85, 88	sbc2ie 3799	. . . . . . 7 ⊢ ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑)
90	89	rabbii 3408	. . . . . 6 ⊢ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑}
91	90	rexeqi 3347	. . . . 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 2808	. . . 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 2794	. . 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 481	. 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 483	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ0)
96		nn0z 12343	. . . . . 6 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℤ)
97		uzid 12597	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ≥‘𝑁))
98		peano2uz 12641	. . . . . 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 2843	. . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑀 ∈ (ℤ≥‘𝑁))
101	100	adantr 481	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ≥‘𝑁))
102		simpr 485	. . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
103		diophrex 40597	. . 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 1370	. 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 2839	1 ⊢ ((𝑁 ∈ ℕ0 ∧ {𝑡 ∈ (ℕ0 ↑m (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑢 ∈ (ℕ0 ↑m (1...𝑁)) ∣ ∃𝑣 ∈ ℕ0 𝜓} ∈ (Dioph‘𝑁))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  {crab 3068  [wsbc 3716   ∪ cun 3885  {csn 4561  ⟨cop 4567   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615  1c1 10872   + caddc 10874  ℕcn 11973  ℕ₀cn0 12233  ℤcz 12319  ℤ_≥cuz 12582  ...cfz 13239  Diophcdioph 40577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-dju 9659  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-hash 14045  df-mzpcl 40545  df-mzp 40546  df-dioph 40578

This theorem is referenced by:  rexfrabdioph  40617  elnn0rabdioph  40625  dvdsrabdioph  40632