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Theorem uzin2 15056
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 12585 . . . 4 :ℤ⟶𝒫 ℤ
2 ffn 6600 . . . 4 (ℤ:ℤ⟶𝒫 ℤ → ℤ Fn ℤ)
31, 2ax-mp 5 . . 3 Fn ℤ
4 fvelrnb 6830 . . 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 6830 . . 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 4139 . . 3 ((ℤ𝑥) = 𝐴 → ((ℤ𝑥) ∩ (ℤ𝑦)) = (𝐴 ∩ (ℤ𝑦)))
98eleq1d 2823 . 2 ((ℤ𝑥) = 𝐴 → (((ℤ𝑥) ∩ (ℤ𝑦)) ∈ ran ℤ ↔ (𝐴 ∩ (ℤ𝑦)) ∈ ran ℤ))
10 ineq2 4140 . . 3 ((ℤ𝑦) = 𝐵 → (𝐴 ∩ (ℤ𝑦)) = (𝐴𝐵))
1110eleq1d 2823 . 2 ((ℤ𝑦) = 𝐵 → ((𝐴 ∩ (ℤ𝑦)) ∈ ran ℤ ↔ (𝐴𝐵) ∈ ran ℤ))
12 uzin 12618 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑥) ∩ (ℤ𝑦)) = (ℤ‘if(𝑥𝑦, 𝑦, 𝑥)))
13 ifcl 4504 . . . . 5 ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℤ)
1413ancoms 459 . . . 4 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑥𝑦, 𝑦, 𝑥) ∈ ℤ)
15 fnfvelrn 6958 . . . 4 ((ℤ Fn ℤ ∧ if(𝑥𝑦, 𝑦, 𝑥) ∈ ℤ) → (ℤ‘if(𝑥𝑦, 𝑦, 𝑥)) ∈ ran ℤ)
163, 14, 15sylancr 587 . . 3 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ‘if(𝑥𝑦, 𝑦, 𝑥)) ∈ ran ℤ)
1712, 16eqeltrd 2839 . 2 ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ𝑥) ∩ (ℤ𝑦)) ∈ ran ℤ)
185, 7, 9, 11, 172gencl 3472 1 ((𝐴 ∈ ran ℤ𝐵 ∈ ran ℤ) → (𝐴𝐵) ∈ ran ℤ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wrex 3065  cin 3886  ifcif 4459  𝒫 cpw 4533   class class class wbr 5074  ran crn 5590   Fn wfn 6428  wf 6429  cfv 6433  cle 11010  cz 12319  cuz 12582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-neg 11208  df-z 12320  df-uz 12583
This theorem is referenced by:  rexanuz  15057  zfbas  23047  heibor1lem  35967
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