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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 14994 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (𝑅↑𝑟0) = ( I ↾ ∪ ∪ 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∪ cuni 4903 I cid 5569 ↾ cres 5674 Rel wrel 5677 (class class class)co 7414 0cc0 11130 ↑𝑟crelexp 14990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-mulcl 11192 ax-i2m1 11198 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-n0 12495 df-relexp 14991 |
This theorem is referenced by: rtrclreclem2 15030 rtrclreclem4 15032 |
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