rexrabdioph - Mathbox for Stefan O'Rear

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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 = (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 2905	. . . . . 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 3404	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓))
3	2	cbvrexv 3148	. . . . . . . . . 10 ⊢ (∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓 ↔ ∃𝑐 ∈ ℕ₀ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)
4	3	anbi2i 726	. . . . . . . . 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 3073	. . . . . . . . 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 266	. . . . . . . 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 786	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑁 ∈ ℕ₀)
8		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)))
9		simplr 788	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑐 ∈ ℕ₀)
10		rexrabdioph.1	. . . . . . . . . . . . . . 15 ⊢ 𝑀 = (𝑁 + 1)
11	10	mapfzcons 36297	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_𝑚 (1...𝑀)))
12	7, 8, 9, 11	syl3anc 1318	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_𝑚 (1...𝑀)))
13	12	adantrr 749	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → (𝑎 ∪ {⟨𝑀, 𝑐⟩}) ∈ (ℕ₀ ↑_𝑚 (1...𝑀)))
14	10	mapfzcons2 36300	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ 𝑐 ∈ ℕ₀) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
15	8, 9, 14	syl2anc 691	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀) = 𝑐)
16	15	eqcomd 2616	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑐 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
17	10	mapfzcons1 36298	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
18	17	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) = 𝑎)
19	18	eqcomd 2616	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
20	19	sbceq1d 3407	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁))) → ([𝑎 / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
21	16, 20	sbceqbid 3409	. . . . . . . . . . . . . 14 ⊢ (((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 218	. . . . . . . . . . . . 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))
23	22	impr 647	. . . . . . . . . . . 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)
24	19	adantrr 749	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑐 ∈ ℕ₀) ∧ (𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓)) → 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
25		fveq1 6102	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏‘𝑀) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩})‘𝑀))
26		reseq1 5311	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑏 ↾ (1...𝑁)) = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)))
27	26	sbceq1d 3407	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → ([(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁)) / 𝑢]𝜓))
28	25, 27	sbceqbid 3409	. . . . . . . . . . . . . 14 ⊢ (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 2620	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑎 ∪ {⟨𝑀, 𝑐⟩}) → (𝑎 = (𝑏 ↾ (1...𝑁)) ↔ 𝑎 = ((𝑎 ∪ {⟨𝑀, 𝑐⟩}) ↾ (1...𝑁))))
30	28, 29	anbi12d 743	. . . . . . . . . . . . 13 ⊢ (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 3282	. . . . . . . . . . . 12 ⊢ (((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 1316	. . . . . . . . . . 11 ⊢ (((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	ex 449	. . . . . . . . . 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)))))
34	33	rexlimdva 3013	. . . . . . . . 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)))))
35		elmapi 7765	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) → 𝑏:(1...𝑀)⟶ℕ₀)
36		nn0p1nn 11209	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ ℕ)
37	10, 36	syl5eqel 2692	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ ℕ)
38		elfz1end 12242	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ ℕ ↔ 𝑀 ∈ (1...𝑀))
39	37, 38	sylib 207	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (1...𝑀))
40		ffvelrn 6265	. . . . . . . . . . . . . 14 ⊢ ((𝑏:(1...𝑀)⟶ℕ₀ ∧ 𝑀 ∈ (1...𝑀)) → (𝑏‘𝑀) ∈ ℕ₀)
41	35, 39, 40	syl2anr 494	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀))) → (𝑏‘𝑀) ∈ ℕ₀)
42	41	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → (𝑏‘𝑀) ∈ ℕ₀)
43		simprr 792	. . . . . . . . . . . . 13 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 = (𝑏 ↾ (1...𝑁)))
44	10	mapfzcons1cl 36299	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) → (𝑏 ↾ (1...𝑁)) ∈ (ℕ₀ ↑_𝑚 (1...𝑁)))
45	44	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((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)))
46	43, 45	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑏 ∈ (ℕ₀ ↑_𝑚 (1...𝑀))) ∧ ([(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓 ∧ 𝑎 = (𝑏 ↾ (1...𝑁)))) → 𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)))
47		simprl 790	. . . . . . . . . . . . 13 ⊢ (((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)
48		dfsbcq 3404	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([𝑎 / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
49	48	sbcbidv 3457	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (𝑏 ↾ (1...𝑁)) → ([(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
50	49	ad2antll 761	. . . . . . . . . . . . 13 ⊢ (((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))
51	47, 50	mpbird 246	. . . . . . . . . . . 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)
52		dfsbcq 3404	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑏‘𝑀) → ([𝑐 / 𝑣][𝑎 / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓))
53	52	anbi2d 736	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑏‘𝑀) → ((𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ [𝑐 / 𝑣][𝑎 / 𝑢]𝜓) ↔ (𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∧ [(𝑏‘𝑀) / 𝑣][𝑎 / 𝑢]𝜓)))
54	53	rspcev 3282	. . . . . . . . . . . 12 ⊢ (((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))
55	42, 46, 51, 54	syl12anc 1316	. . . . . . . . . . 11 ⊢ (((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	55	ex 449	. . . . . . . . . 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)))
57	56	rexlimdva 3013	. . . . . . . . 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)))
58	34, 57	impbid 201	. . . . . . . 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)))))
59	6, 58	syl5bb 271	. . . . . . 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)))))
60	59	abbidv 2728	. . . . . 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)))})
61	1, 60	syl5eq 2656	. . . . 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)))})
62		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑢(ℕ₀ ↑_𝑚 (1...𝑁))
63		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑎(ℕ₀ ↑_𝑚 (1...𝑁))
64		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑎∃𝑣 ∈ ℕ₀ 𝜓
65		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑢ℕ₀
66		nfcv 2751	. . . . . . . 8 ⊢ Ⅎ𝑢𝑏
67		nfsbc1v 3422	. . . . . . . 8 ⊢ Ⅎ𝑢[𝑎 / 𝑢]𝜓
68	66, 67	nfsbc 3424	. . . . . . 7 ⊢ Ⅎ𝑢[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
69	65, 68	nfrex 2990	. . . . . 6 ⊢ Ⅎ𝑢∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓
70		sbceq1a 3413	. . . . . . . 8 ⊢ (𝑢 = 𝑎 → (𝜓 ↔ [𝑎 / 𝑢]𝜓))
71	70	rexbidv 3034	. . . . . . 7 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓))
72		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑏[𝑎 / 𝑢]𝜓
73		nfsbc1v 3422	. . . . . . . 8 ⊢ Ⅎ𝑣[𝑏 / 𝑣][𝑎 / 𝑢]𝜓
74		sbceq1a 3413	. . . . . . . 8 ⊢ (𝑣 = 𝑏 → ([𝑎 / 𝑢]𝜓 ↔ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
75	72, 73, 74	cbvrex 3144	. . . . . . 7 ⊢ (∃𝑣 ∈ ℕ₀ [𝑎 / 𝑢]𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓)
76	71, 75	syl6bb 275	. . . . . 6 ⊢ (𝑢 = 𝑎 → (∃𝑣 ∈ ℕ₀ 𝜓 ↔ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓))
77	62, 63, 64, 69, 76	cbvrab 3171	. . . . 5 ⊢ {𝑢 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∣ ∃𝑏 ∈ ℕ₀ [𝑏 / 𝑣][𝑎 / 𝑢]𝜓}
78		fveq1 6102	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → (𝑡‘𝑀) = (𝑏‘𝑀))
79		reseq1 5311	. . . . . . . . 9 ⊢ (𝑡 = 𝑏 → (𝑡 ↾ (1...𝑁)) = (𝑏 ↾ (1...𝑁)))
80	79	sbceq1d 3407	. . . . . . . 8 ⊢ (𝑡 = 𝑏 → ([(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
81	78, 80	sbceqbid 3409	. . . . . . 7 ⊢ (𝑡 = 𝑏 → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ [(𝑏‘𝑀) / 𝑣][(𝑏 ↾ (1...𝑁)) / 𝑢]𝜓))
82	81	rexrab 3337	. . . . . 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))))
83	82	abbii 2726	. . . . 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)))}
84	61, 77, 83	3eqtr4g 2669	. . . 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))})
85		fvex 6113	. . . . . . . . 9 ⊢ (𝑡‘𝑀) ∈ V
86		vex 3176	. . . . . . . . . 10 ⊢ 𝑡 ∈ V
87	86	resex 5363	. . . . . . . . 9 ⊢ (𝑡 ↾ (1...𝑁)) ∈ V
88		rexrabdioph.2	. . . . . . . . . 10 ⊢ (𝑣 = (𝑡‘𝑀) → (𝜓 ↔ 𝜒))
89		rexrabdioph.3	. . . . . . . . . 10 ⊢ (𝑢 = (𝑡 ↾ (1...𝑁)) → (𝜒 ↔ 𝜑))
90	88, 89	sylan9bb 732	. . . . . . . . 9 ⊢ ((𝑣 = (𝑡‘𝑀) ∧ 𝑢 = (𝑡 ↾ (1...𝑁))) → (𝜓 ↔ 𝜑))
91	85, 87, 90	sbc2ie 3472	. . . . . . . 8 ⊢ ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑)
92	91	a1i 11	. . . . . . 7 ⊢ (𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) → ([(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓 ↔ 𝜑))
93	92	rabbiia 3161	. . . . . 6 ⊢ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ [(𝑡‘𝑀) / 𝑣][(𝑡 ↾ (1...𝑁)) / 𝑢]𝜓} = {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑}
94	93	rexeqi 3120	. . . . 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)))
95	94	abbii 2726	. . . 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))}
96	84, 95	syl6eq 2660	. . 3 ⊢ (𝑁 ∈ ℕ₀ → {𝑢 ∈ (ℕ₀ ↑_𝑚 (1...𝑁)) ∣ ∃𝑣 ∈ ℕ₀ 𝜓} = {𝑎 ∣ ∃𝑏 ∈ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑}𝑎 = (𝑏 ↾ (1...𝑁))})
97	96	adantr 480	. 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))})
98		simpl 472	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑁 ∈ ℕ₀)
99		nn0z 11277	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
100		uzid 11578	. . . . . 6 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ_≥‘𝑁))
101		peano2uz 11617	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑁) → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
102	99, 100, 101	3syl 18	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (𝑁 + 1) ∈ (ℤ_≥‘𝑁))
103	10, 102	syl5eqel 2692	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → 𝑀 ∈ (ℤ_≥‘𝑁))
104	103	adantr 480	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑁))
105		simpr 476	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀)) → {𝑡 ∈ (ℕ₀ ↑_𝑚 (1...𝑀)) ∣ 𝜑} ∈ (Dioph‘𝑀))
106		diophrex 36357	. . 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))
107	98, 104, 105, 106	syl3anc 1318	. 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))
108	97, 107	eqeltrd 2688	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 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

W3C validator