Theorem uzin2 15363

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 12855	. . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ
2		ffn 6706	. . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ)
3	1, 2	ax-mp 5	. . 3 ⊢ ℤ≥ Fn ℤ
4		fvelrnb 6939	. . 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 6939	. . 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 4188	. . 3 ⊢ ((ℤ≥‘𝑥) = 𝐴 → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ (ℤ≥‘𝑦)))
9	8	eleq1d 2819	. 2 ⊢ ((ℤ≥‘𝑥) = 𝐴 → (((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥))
10		ineq2 4189	. . 3 ⊢ ((ℤ≥‘𝑦) = 𝐵 → (𝐴 ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ 𝐵))
11	10	eleq1d 2819	. 2 ⊢ ((ℤ≥‘𝑦) = 𝐵 → ((𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ 𝐵) ∈ ran ℤ≥))
12		uzin 12892	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)))
13		ifcl 4546	. . . . 5 ⊢ ((𝑦 ∈ ℤ ∧ 𝑥 ∈ ℤ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ)
14	13	ancoms 458	. . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ)
15		fnfvelrn 7070	. . . 4 ⊢ ((ℤ≥ Fn ℤ ∧ if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥)
16	3, 14, 15	sylancr 587	. . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥)
17	12, 16	eqeltrd 2834	. 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥)
18	5, 7, 9, 11, 17	2gencl 3503	1 ⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   ∩ cin 3925  ifcif 4500  𝒫 cpw 4575   class class class wbr 5119  ran crn 5655   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531   ≤ cle 11270  ℤcz 12588  ℤ_≥cuz 12852

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-pre-lttri 11203  ax-pre-lttrn 11204

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-neg 11469  df-z 12589  df-uz 12853

This theorem is referenced by:  rexanuz  15364  zfbas  23834  heibor1lem  37833