Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 12910 | . . . 4 ⊢ (𝑚...(𝑛 − 1)) ⊆ ℤ | |
2 | ovex 7189 | . . . . 5 ⊢ (𝑚...(𝑛 − 1)) ∈ V | |
3 | 2 | elpw 4543 | . . . 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 13035 | . . 3 ⊢ ..^ = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ (𝑚...(𝑛 − 1))) | |
7 | 6 | fmpo 7766 | . 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 2114 ∀wral 3138 ⊆ wss 3936 𝒫 cpw 4539 × cxp 5553 ⟶wf 6351 (class class class)co 7156 1c1 10538 − cmin 10870 ℤcz 11982 ...cfz 12893 ..^cfzo 13034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-neg 10873 df-z 11983 df-uz 12245 df-fz 12894 df-fzo 13035 |
This theorem is referenced by: elfzoel1 13037 elfzoel2 13038 elfzoelz 13039 fzoval 13040 fzofi 13343 |
Copyright terms: Public domain | W3C validator |