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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 13529 | . . . 4 ⊢ (𝑚...(𝑛 − 1)) ⊆ ℤ | |
2 | ovex 7447 | . . . . 5 ⊢ (𝑚...(𝑛 − 1)) ∈ V | |
3 | 2 | elpw 4602 | . . . 4 ⊢ ((𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ ↔ (𝑚...(𝑛 − 1)) ⊆ ℤ) |
4 | 1, 3 | mpbir 230 | . . 3 ⊢ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
5 | 4 | rgen2w 3062 | . 2 ⊢ ∀𝑚 ∈ ℤ ∀𝑛 ∈ ℤ (𝑚...(𝑛 − 1)) ∈ 𝒫 ℤ |
6 | df-fzo 13654 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
7 | 6 | fmpo 8066 | . 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 2099 ∀wral 3057 ⊆ wss 3945 𝒫 cpw 4598 × cxp 5670 ⟶wf 6538 (class class class)co 7414 1c1 11133 − cmin 11468 ℤcz 12582 ...cfz 13510 ..^cfzo 13653 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-neg 11471 df-z 12583 df-uz 12847 df-fz 13511 df-fzo 13654 |
This theorem is referenced by: elfzoel1 13656 elfzoel2 13657 elfzoelz 13658 fzoval 13659 fzofi 13965 |
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