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Mirrors > Home > MPE Home > Th. List > relexp0rel | Structured version Visualization version GIF version |
Description: The exponentiation of a class to zero is a relation. (Contributed by RP, 23-May-2020.) |
Ref | Expression |
---|---|
relexp0rel | ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relres 5875 | . 2 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅)) | |
2 | relexp0g 14369 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | |
3 | 2 | releqd 5646 | . 2 ⊢ (𝑅 ∈ 𝑉 → (Rel (𝑅↑𝑟0) ↔ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
4 | 1, 3 | mpbiri 259 | 1 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∪ cun 3931 I cid 5452 dom cdm 5548 ran crn 5549 ↾ cres 5550 Rel wrel 5553 (class class class)co 7145 0cc0 10525 ↑𝑟crelexp 14367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-mulcl 10587 ax-i2m1 10593 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-n0 11886 df-relexp 14368 |
This theorem is referenced by: relexprelg 14385 relexpaddg 14400 |
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