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Mirrors > Home > MPE Home > Th. List > fzof | Structured version Visualization version GIF version |
Description: Functionality of the half-open integer set function. (Contributed by Stefan O'Rear, 14-Aug-2015.) |
Ref | Expression |
---|---|
fzof | ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzssz 12903 | . . . 4 ⊢ (𝑚...(𝑛 − 1)) ⊆ ℤ | |
2 | ovex 7183 | . . . . 5 ⊢ (𝑚...(𝑛 − 1)) ∈ V | |
3 | 2 | elpw 4545 | . . . 4 ⊢ ((𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ (𝑚...(𝑛 − 1)) ⊆ ℤ) |
4 | 1, 3 | mpbir 233 | . . 3 ⊢ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
5 | 4 | rgen2w 3151 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
6 | df-fzo 13028 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
7 | 6 | fmpo 7760 | . 2 ⊢ (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ ..^:(ℤ × ℤ)⟶𝒫 ℤ) |
8 | 5, 7 | mpbi 232 | 1 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 ∀wral 3138 ⊆ wss 3935 𝒫 cpw 4538 × cxp 5547 ⟶wf 6345 (class class class)co 7150 1c1 10532 − cmin 10864 ℤcz 11975 ...cfz 12886 ..^cfzo 13027 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-neg 10867 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 |
This theorem is referenced by: elfzoel1 13030 elfzoel2 13031 elfzoelz 13032 fzoval 13033 fzofi 13336 |
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