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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 13563 | . . . 4 ⊢ (𝑚...(𝑛 − 1)) ⊆ ℤ | |
2 | ovex 7464 | . . . . 5 ⊢ (𝑚...(𝑛 − 1)) ∈ V | |
3 | 2 | elpw 4609 | . . . 4 ⊢ ((𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ (𝑚...(𝑛 − 1)) ⊆ ℤ) |
4 | 1, 3 | mpbir 231 | . . 3 ⊢ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
5 | 4 | rgen2w 3064 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
6 | df-fzo 13692 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
7 | 6 | fmpo 8092 | . 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 2106 ∀wral 3059 ⊆ wss 3963 𝒫 cpw 4605 × cxp 5687 ⟶wf 6559 (class class class)co 7431 1c1 11154 − cmin 11490 ℤcz 12611 ...cfz 13544 ..^cfzo 13691 |
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-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 |
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-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-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 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-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-neg 11493 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 |
This theorem is referenced by: elfzoel1 13694 elfzoel2 13695 elfzoelz 13696 fzoval 13697 fzofi 14012 |
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