![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > relcoi1 | Structured version Visualization version GIF version |
Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 8-May-2011.) (Proof shortened by OpenAI, 3-Jul-2020.) |
Ref | Expression |
---|---|
relcoi1 | ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coires1 6267 | . 2 ⊢ (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ↾ ∪ ∪ 𝑅) | |
2 | relresfld 6279 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | |
3 | 1, 2 | eqtrid 2778 | 1 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∪ cuni 4907 I cid 5571 ↾ cres 5676 ∘ ccom 5678 Rel wrel 5679 |
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-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3466 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 |
This theorem is referenced by: relexpsucl 15030 tsrdir 18623 |
Copyright terms: Public domain | W3C validator |