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Theorem uzin2 13932
 Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.)
Assertion
Ref Expression
uzin2 ((𝐴 ∈ ran ℤ𝐵 ∈ ran ℤ) → (𝐴𝐵) ∈ ran ℤ)

Proof of Theorem uzin2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uzf 11566 . . . 4 :ℤ⟶𝒫 ℤ
2 ffn 5958 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
31, 2ax-mp 5 . . 3 Fn ℤ
4 fvelrnb 6153 . . 3 (ℤ Fn ℤ → (𝐴 ∈ ran ℤ ↔ ∃𝑥 ∈ ℤ (ℤ𝑥) = 𝐴))
53, 4ax-mp 5 . 2 (𝐴 ∈ ran ℤ ↔ ∃𝑥 ∈ ℤ (ℤ𝑥) = 𝐴)
6 fvelrnb 6153 . . 3 (ℤ Fn ℤ → (𝐵 ∈ ran ℤ ↔ ∃𝑦 ∈ ℤ (ℤ𝑦) = 𝐵))
73, 6ax-mp 5 . 2 (𝐵 ∈ ran ℤ ↔ ∃𝑦 ∈ ℤ (ℤ𝑦) = 𝐵)
8 ineq1 3769 . . 3 ((ℤ𝑥) = 𝐴 → ((ℤ𝑥) ∩ (ℤ𝑦)) = (𝐴 ∩ (ℤ𝑦)))
98eleq1d 2672 . 2 ((ℤ𝑥) = 𝐴 → (((ℤ𝑥) ∩ (ℤ𝑦)) ∈ ran ℤ ↔ (𝐴 ∩ (ℤ𝑦)) ∈ ran ℤ))
10 ineq2 3770 . . 3 ((ℤ𝑦) = 𝐵 → (𝐴 ∩ (ℤ𝑦)) = (𝐴𝐵))
1110eleq1d 2672 . 2 ((ℤ𝑦) = 𝐵 → ((𝐴 ∩ (ℤ𝑦)) ∈ ran ℤ ↔ (𝐴𝐵) ∈ ran ℤ))
12 uzin 11596 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑥) ∩ (ℤ𝑦)) = (ℤ‘if(𝑥𝑦, 𝑦, 𝑥)))
13 ifcl 4080 . . . . 5 ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℤ)
1413ancoms 468 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℤ)
15 fnfvelrn 6264 . . . 4 ((ℤ Fn ℤ ∧ if(𝑥𝑦, 𝑦, 𝑥) ∈ ℤ) → (ℤ‘if(𝑥𝑦, 𝑦, 𝑥)) ∈ ran ℤ)
163, 14, 15sylancr 694 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ‘if(𝑥𝑦, 𝑦, 𝑥)) ∈ ran ℤ)
1712, 16eqeltrd 2688 . 2 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑥) ∩ (ℤ𝑦)) ∈ ran ℤ)
185, 7, 9, 11, 172gencl 3209 1 ((𝐴 ∈ ran ℤ𝐵 ∈ ran ℤ) → (𝐴𝐵) ∈ ran ℤ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   ∩ cin 3539  ifcif 4036  𝒫 cpw 4108   class class class wbr 4583  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804   ≤ cle 9954  ℤcz 11254  ℤ≥cuz 11563 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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 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-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-neg 10148  df-z 11255  df-uz 11564 This theorem is referenced by:  rexanuz  13933  zfbas  21510  heibor1lem  32778
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