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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 5868 | . 2 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅)) | |
2 | relexp0g 14366 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | |
3 | 2 | releqd 5639 | . 2 ⊢ (𝑅 ∈ 𝑉 → (Rel (𝑅↑𝑟0) ↔ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
4 | 1, 3 | mpbiri 260 | 1 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∪ cun 3922 I cid 5445 dom cdm 5541 ran crn 5542 ↾ cres 5543 Rel wrel 5546 (class class class)co 7142 0cc0 10523 ↑𝑟crelexp 14364 |
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 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-mulcl 10585 ax-i2m1 10591 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-iota 6300 df-fun 6343 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-n0 11885 df-relexp 14365 |
This theorem is referenced by: relexprelg 14382 relexpaddg 14397 |
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