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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 13444 | . . . 4 ⊢ (𝑚...(𝑛 − 1)) ⊆ ℤ | |
2 | ovex 7391 | . . . . 5 ⊢ (𝑚...(𝑛 − 1)) ∈ V | |
3 | 2 | elpw 4565 | . . . 4 ⊢ ((𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ (𝑚...(𝑛 − 1)) ⊆ ℤ) |
4 | 1, 3 | mpbir 230 | . . 3 ⊢ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
5 | 4 | rgen2w 3070 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
6 | df-fzo 13569 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
7 | 6 | fmpo 8001 | . 2 ⊢ (∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ ..^:(ℤ × ℤ)⟶𝒫 ℤ) |
8 | 5, 7 | mpbi 229 | 1 ⊢ ..^:(ℤ × ℤ)⟶𝒫 ℤ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∀wral 3065 ⊆ wss 3911 𝒫 cpw 4561 × cxp 5632 ⟶wf 6493 (class class class)co 7358 1c1 11053 − cmin 11386 ℤcz 12500 ...cfz 13425 ..^cfzo 13568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-neg 11389 df-z 12501 df-uz 12765 df-fz 13426 df-fzo 13569 |
This theorem is referenced by: elfzoel1 13571 elfzoel2 13572 elfzoelz 13573 fzoval 13574 fzofi 13880 |
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