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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 5426 | . 2 ⊢ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅)) | |
2 | relexp0g 13762 | . . 3 ⊢ (𝑅 ∈ 𝑉 → (𝑅↑𝑟0) = ( I ↾ (dom 𝑅 ∪ ran 𝑅))) | |
3 | 2 | releqd 5203 | . 2 ⊢ (𝑅 ∈ 𝑉 → (Rel (𝑅↑𝑟0) ↔ Rel ( I ↾ (dom 𝑅 ∪ ran 𝑅)))) |
4 | 1, 3 | mpbiri 248 | 1 ⊢ (𝑅 ∈ 𝑉 → Rel (𝑅↑𝑟0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∪ cun 3572 I cid 5023 dom cdm 5114 ran crn 5115 ↾ cres 5116 Rel wrel 5119 (class class class)co 6650 0cc0 9936 ↑𝑟crelexp 13760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-i2m1 10004 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-n0 11293 df-relexp 13761 |
This theorem is referenced by: relexprelg 13778 relexpaddg 13793 |
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