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Mirrors > Home > MPE Home > Th. List > relexp0d | Structured version Visualization version GIF version |
Description: A relation composed zero times is the (restricted) identity. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 12-Jul-2024.) |
Ref | Expression |
---|---|
relexp0d.1 | ⊢ (𝜑 → Rel 𝑅) |
relexp0d.2 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
Ref | Expression |
---|---|
relexp0d | ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relexp0d.2 | . 2 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
2 | relexp0d.1 | . 2 ⊢ (𝜑 → Rel 𝑅) | |
3 | relexp0 14917 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | |
4 | 1, 2, 3 | syl2anc 585 | 1 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∪ cuni 4869 I cid 5534 ↾ cres 5639 Rel wrel 5642 (class class class)co 7361 0cc0 11059 ↑𝑟crelexp 14913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-mulcl 11121 ax-i2m1 11127 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-iota 6452 df-fun 6502 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-n0 12422 df-relexp 14914 |
This theorem is referenced by: rtrclreclem2 14953 rtrclreclem4 14955 |
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