![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uzin2 | Structured version Visualization version GIF version |
Description: The upper integers are closed under intersection. (Contributed by Mario Carneiro, 24-Dec-2013.) |
Ref | Expression |
---|---|
uzin2 | ⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzf 12879 | . . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | ffn 6737 | . . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ℤ≥ Fn ℤ |
4 | fvelrnb 6969 | . . 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 6969 | . . 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 4221 | . . 3 ⊢ ((ℤ≥‘𝑥) = 𝐴 → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ (ℤ≥‘𝑦))) | |
9 | 8 | eleq1d 2824 | . 2 ⊢ ((ℤ≥‘𝑥) = 𝐴 → (((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥)) |
10 | ineq2 4222 | . . 3 ⊢ ((ℤ≥‘𝑦) = 𝐵 → (𝐴 ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ 𝐵)) | |
11 | 10 | eleq1d 2824 | . 2 ⊢ ((ℤ≥‘𝑦) = 𝐵 → ((𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ 𝐵) ∈ ran ℤ≥)) |
12 | uzin 12916 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥))) | |
13 | ifcl 4576 | . . . . 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 7100 | . . . 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 2839 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥) |
18 | 5, 7, 9, 11, 17 | 2gencl 3522 | 1 ⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ∩ cin 3962 ifcif 4531 𝒫 cpw 4605 class class class wbr 5148 ran crn 5690 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 ≤ cle 11294 ℤcz 12611 ℤ≥cuz 12876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-z 12612 df-uz 12877 |
This theorem is referenced by: rexanuz 15381 zfbas 23920 heibor1lem 37796 |
Copyright terms: Public domain | W3C validator |