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| 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 13566 | . . . 4 ⊢ (𝑚...(𝑛 − 1)) ⊆ ℤ | |
| 2 | ovex 7464 | . . . . 5 ⊢ (𝑚...(𝑛 − 1)) ∈ V | |
| 3 | 2 | elpw 4604 | . . . 4 ⊢ ((𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ (𝑚...(𝑛 − 1)) ⊆ ℤ) | 
| 4 | 1, 3 | mpbir 231 | . . 3 ⊢ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ | 
| 5 | 4 | rgen2w 3066 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ | 
| 6 | df-fzo 13695 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
| 7 | 6 | fmpo 8093 | . 2 ⊢ (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ ..^:(ℤ × ℤ)⟶𝒫 ℤ) | 
| 8 | 5, 7 | mpbi 230 | 1 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∈ wcel 2108 ∀wral 3061 ⊆ wss 3951 𝒫 cpw 4600 × cxp 5683 ⟶wf 6557 (class class class)co 7431 1c1 11156 − cmin 11492 ℤcz 12613 ...cfz 13547 ..^cfzo 13694 | 
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-neg 11495 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 | 
| This theorem is referenced by: elfzoel1 13697 elfzoel2 13698 elfzoelz 13699 fzoval 13700 fzofi 14015 | 
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