Theorem uzin2 10483

Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)

Assertion

Ref	Expression
uzin2	⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥)

Proof of Theorem uzin2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzf 9085	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 5176	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
3	1, 2	ax-mp 7	. . 3 ⊢ ℤ≥ Fn ℤ
4		fvelrnb 5367	. . 3 ⊢ (ℤ≥ Fn ℤ → (𝐴 ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = 𝐴))
5	3, 4	ax-mp 7	. 2 ⊢ (𝐴 ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = 𝐴)
6		fvelrnb 5367	. . 3 ⊢ (ℤ≥ Fn ℤ → (𝐵 ∈ ran ℤ≥ ↔ ∃𝑦 ∈ ℤ (ℤ≥‘𝑦) = 𝐵))
7	3, 6	ax-mp 7	. 2 ⊢ (𝐵 ∈ ran ℤ≥ ↔ ∃𝑦 ∈ ℤ (ℤ≥‘𝑦) = 𝐵)
8		ineq1 3197	. . 3 ⊢ ((ℤ≥‘𝑥) = 𝐴 → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ (ℤ≥‘𝑦)))
9	8	eleq1d 2157	. 2 ⊢ ((ℤ≥‘𝑥) = 𝐴 → (((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥))
10		ineq2 3198	. . 3 ⊢ ((ℤ≥‘𝑦) = 𝐵 → (𝐴 ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ 𝐵))
11	10	eleq1d 2157	. 2 ⊢ ((ℤ≥‘𝑦) = 𝐵 → ((𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ 𝐵) ∈ ran ℤ≥))
12		uzin 9114	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)))
13		simpr 109	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
14		simpl 108	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
15		zdcle 8886	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 ≤ 𝑦)
16	13, 14, 15	ifcldcd 3432	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ)
17		fnfvelrn 5447	. . . 4 ⊢ ((ℤ≥ Fn ℤ ∧ if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥)
18	3, 16, 17	sylancr 406	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥)
19	12, 18	eqeltrd 2165	. 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥)
20	5, 7, 9, 11, 19	2gencl 2655	1 ⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1290   ∈ wcel 1439  ∃wrex 2361   ∩ cin 3001  ifcif 3399  𝒫 cpw 3435   class class class wbr 3853  ran crn 4455   Fn wfn 5025  ⟶wf 5026  ‘cfv 5030   ≤ cle 7586  ℤcz 8813  ℤ_≥cuz 9082

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-1re 7502  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-addcom 7508  ax-addass 7510  ax-distr 7512  ax-i2m1 7513  ax-0lt1 7514  ax-0id 7516  ax-rnegex 7517  ax-cnre 7519  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522  ax-pre-apti 7523  ax-pre-ltadd 7524

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-if 3400  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-br 3854  df-opab 3908  df-mpt 3909  df-id 4131  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-fv 5038  df-riota 5624  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-sub 7718  df-neg 7719  df-inn 8486  df-n0 8737  df-z 8814  df-uz 9083

This theorem is referenced by:  rexanuz  10484