Theorem uzin2 11753

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 9924	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 5533	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
3	1, 2	ax-mp 5	. . 3 ⊢ ℤ≥ Fn ℤ
4		fvelrnb 5750	. . 3 ⊢ (ℤ≥ Fn ℤ → (𝐴 ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = 𝐴))
5	3, 4	ax-mp 5	. 2 ⊢ (𝐴 ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = 𝐴)
6		fvelrnb 5750	. . 3 ⊢ (ℤ≥ Fn ℤ → (𝐵 ∈ ran ℤ≥ ↔ ∃𝑦 ∈ ℤ (ℤ≥‘𝑦) = 𝐵))
7	3, 6	ax-mp 5	. 2 ⊢ (𝐵 ∈ ran ℤ≥ ↔ ∃𝑦 ∈ ℤ (ℤ≥‘𝑦) = 𝐵)
8		ineq1 3425	. . 3 ⊢ ((ℤ≥‘𝑥) = 𝐴 → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ (ℤ≥‘𝑦)))
9	8	eleq1d 2307	. 2 ⊢ ((ℤ≥‘𝑥) = 𝐴 → (((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥))
10		ineq2 3426	. . 3 ⊢ ((ℤ≥‘𝑦) = 𝐵 → (𝐴 ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ 𝐵))
11	10	eleq1d 2307	. 2 ⊢ ((ℤ≥‘𝑦) = 𝐵 → ((𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ 𝐵) ∈ ran ℤ≥))
12		uzin 9955	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)))
13		simpr 110	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ)
14		simpl 109	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ)
15		zdcle 9721	. . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 ≤ 𝑦)
16	13, 14, 15	ifcldcd 3678	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ)
17		fnfvelrn 5840	. . . 4 ⊢ ((ℤ≥ Fn ℤ ∧ if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥)
18	3, 16, 17	sylancr 418	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥)
19	12, 18	eqeltrd 2315	. 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥)
20	5, 7, 9, 11, 19	2gencl 2855	1 ⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ∩ cin 3219  ifcif 3638  𝒫 cpw 3688   class class class wbr 4130  ran crn 4775   Fn wfn 5372  ⟶wf 5373  ‘cfv 5377   ≤ cle 8361  ℤcz 9644  ℤ_≥cuz 9921

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922

This theorem is used by:  rexanuz  11754