Theorem fzopth 10061

Description: A finite set of sequential integers can represent an ordered pair. (Contributed by NM, 31-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
fzopth	⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Proof of Theorem fzopth

Step	Hyp	Ref	Expression
1		eluzfz1 10031	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
2	1	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝑀...𝑁))
3		simpr 110	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀...𝑁) = (𝐽...𝐾))
4	2, 3	eleqtrd 2256	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ (𝐽...𝐾))
5		elfzuz 10021	. . . . . . 7 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝑀 ∈ (ℤ≥‘𝐽))
6		uzss 9548	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘𝐽) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
7	4, 5, 6	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) ⊆ (ℤ≥‘𝐽))
8		elfzuz2 10029	. . . . . . . . 9 ⊢ (𝑀 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝐽))
9		eluzfz1 10031	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐽 ∈ (𝐽...𝐾))
10	4, 8, 9	3syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝐽...𝐾))
11	10, 3	eleqtrrd 2257	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐽 ∈ (𝑀...𝑁))
12		elfzuz 10021	. . . . . . 7 ⊢ (𝐽 ∈ (𝑀...𝑁) → 𝐽 ∈ (ℤ≥‘𝑀))
13		uzss 9548	. . . . . . 7 ⊢ (𝐽 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
14	11, 12, 13	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐽) ⊆ (ℤ≥‘𝑀))
15	7, 14	eqssd 3173	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑀) = (ℤ≥‘𝐽))
16		eluzel2 9533	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
17	16	adantr 276	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 ∈ ℤ)
18		uz11 9550	. . . . . 6 ⊢ (𝑀 ∈ ℤ → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
19	17, 18	syl 14	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑀) = (ℤ≥‘𝐽) ↔ 𝑀 = 𝐽))
20	15, 19	mpbid 147	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑀 = 𝐽)
21		eluzfz2 10032	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝐽) → 𝐾 ∈ (𝐽...𝐾))
22	4, 8, 21	3syl 17	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝐽...𝐾))
23	22, 3	eleqtrrd 2257	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝐾 ∈ (𝑀...𝑁))
24		elfzuz3 10022	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
25		uzss 9548	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
26	23, 24, 25	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) ⊆ (ℤ≥‘𝐾))
27		eluzfz2 10032	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
28	27	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝑀...𝑁))
29	28, 3	eleqtrd 2256	. . . . . . 7 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ (𝐽...𝐾))
30		elfzuz3 10022	. . . . . . 7 ⊢ (𝑁 ∈ (𝐽...𝐾) → 𝐾 ∈ (ℤ≥‘𝑁))
31		uzss 9548	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑁) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
32	29, 30, 31	3syl 17	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝐾) ⊆ (ℤ≥‘𝑁))
33	26, 32	eqssd 3173	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (ℤ≥‘𝑁) = (ℤ≥‘𝐾))
34		eluzelz 9537	. . . . . . 7 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
35	34	adantr 276	. . . . . 6 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 ∈ ℤ)
36		uz11 9550	. . . . . 6 ⊢ (𝑁 ∈ ℤ → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
37	35, 36	syl 14	. . . . 5 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → ((ℤ≥‘𝑁) = (ℤ≥‘𝐾) ↔ 𝑁 = 𝐾))
38	33, 37	mpbid 147	. . . 4 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → 𝑁 = 𝐾)
39	20, 38	jca 306	. . 3 ⊢ ((𝑁 ∈ (ℤ≥‘𝑀) ∧ (𝑀...𝑁) = (𝐽...𝐾)) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾))
40	39	ex 115	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) → (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))
41		oveq12 5884	. 2 ⊢ ((𝑀 = 𝐽 ∧ 𝑁 = 𝐾) → (𝑀...𝑁) = (𝐽...𝐾))
42	40, 41	impbid1 142	1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝐽...𝐾) ↔ (𝑀 = 𝐽 ∧ 𝑁 = 𝐾)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ⊆ wss 3130  ‘cfv 5217  (class class class)co 5875  ℤcz 9253  ℤ_≥cuz 9528  ...cfz 10008

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-apti 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-neg 8131  df-z 9254  df-uz 9529  df-fz 10009

This theorem is referenced by:  fz0to4untppr  10124  2ffzeq  10141