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Mirrors > Home > ILE Home > Th. List > uzin2 | 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 8703 | . . . 4 ⊢ ℤ≥:ℤ⟶𝒫 ℤ | |
2 | ffn 5077 | . . . 4 ⊢ (ℤ≥:ℤ⟶𝒫 ℤ → ℤ≥ Fn ℤ) | |
3 | 1, 2 | ax-mp 7 | . . 3 ⊢ ℤ≥ Fn ℤ |
4 | fvelrnb 5253 | . . 3 ⊢ (ℤ≥ Fn ℤ → (𝐴 ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = 𝐴)) | |
5 | 3, 4 | ax-mp 7 | . 2 ⊢ (𝐴 ∈ ran ℤ≥ ↔ ∃𝑥 ∈ ℤ (ℤ≥‘𝑥) = 𝐴) |
6 | fvelrnb 5253 | . . 3 ⊢ (ℤ≥ Fn ℤ → (𝐵 ∈ ran ℤ≥ ↔ ∃𝑦 ∈ ℤ (ℤ≥‘𝑦) = 𝐵)) | |
7 | 3, 6 | ax-mp 7 | . 2 ⊢ (𝐵 ∈ ran ℤ≥ ↔ ∃𝑦 ∈ ℤ (ℤ≥‘𝑦) = 𝐵) |
8 | ineq1 3167 | . . 3 ⊢ ((ℤ≥‘𝑥) = 𝐴 → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ (ℤ≥‘𝑦))) | |
9 | 8 | eleq1d 2148 | . 2 ⊢ ((ℤ≥‘𝑥) = 𝐴 → (((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥)) |
10 | ineq2 3168 | . . 3 ⊢ ((ℤ≥‘𝑦) = 𝐵 → (𝐴 ∩ (ℤ≥‘𝑦)) = (𝐴 ∩ 𝐵)) | |
11 | 10 | eleq1d 2148 | . 2 ⊢ ((ℤ≥‘𝑦) = 𝐵 → ((𝐴 ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥ ↔ (𝐴 ∩ 𝐵) ∈ ran ℤ≥)) |
12 | uzin 8732 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) = (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥))) | |
13 | simpr 108 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ ℤ) | |
14 | simpl 107 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → 𝑥 ∈ ℤ) | |
15 | zdcle 8505 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → DECID 𝑥 ≤ 𝑦) | |
16 | 13, 14, 15 | ifcldcd 3389 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ) |
17 | fnfvelrn 5331 | . . . 4 ⊢ ((ℤ≥ Fn ℤ ∧ if(𝑥 ≤ 𝑦, 𝑦, 𝑥) ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥) | |
18 | 3, 16, 17 | sylancr 405 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (ℤ≥‘if(𝑥 ≤ 𝑦, 𝑦, 𝑥)) ∈ ran ℤ≥) |
19 | 12, 18 | eqeltrd 2156 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((ℤ≥‘𝑥) ∩ (ℤ≥‘𝑦)) ∈ ran ℤ≥) |
20 | 5, 7, 9, 11, 19 | 2gencl 2633 | 1 ⊢ ((𝐴 ∈ ran ℤ≥ ∧ 𝐵 ∈ ran ℤ≥) → (𝐴 ∩ 𝐵) ∈ ran ℤ≥) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 ∃wrex 2350 ∩ cin 2973 ifcif 3359 𝒫 cpw 3390 class class class wbr 3793 ran crn 4372 Fn wfn 4927 ⟶wf 4928 ‘cfv 4932 ≤ cle 7216 ℤcz 8432 ℤ≥cuz 8700 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2064 ax-sep 3904 ax-pow 3956 ax-pr 3972 ax-un 4196 ax-setind 4288 ax-cnex 7129 ax-resscn 7130 ax-1cn 7131 ax-1re 7132 ax-icn 7133 ax-addcl 7134 ax-addrcl 7135 ax-mulcl 7136 ax-addcom 7138 ax-addass 7140 ax-distr 7142 ax-i2m1 7143 ax-0lt1 7144 ax-0id 7146 ax-rnegex 7147 ax-cnre 7149 ax-pre-ltirr 7150 ax-pre-ltwlin 7151 ax-pre-lttrn 7152 ax-pre-apti 7153 ax-pre-ltadd 7154 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-if 3360 df-pw 3392 df-sn 3412 df-pr 3413 df-op 3415 df-uni 3610 df-int 3645 df-br 3794 df-opab 3848 df-mpt 3849 df-id 4056 df-xp 4377 df-rel 4378 df-cnv 4379 df-co 4380 df-dm 4381 df-rn 4382 df-res 4383 df-ima 4384 df-iota 4897 df-fun 4934 df-fn 4935 df-f 4936 df-fv 4940 df-riota 5499 df-ov 5546 df-oprab 5547 df-mpt2 5548 df-pnf 7217 df-mnf 7218 df-xr 7219 df-ltxr 7220 df-le 7221 df-sub 7348 df-neg 7349 df-inn 8107 df-n0 8356 df-z 8433 df-uz 8701 |
This theorem is referenced by: rexanuz 10012 |
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