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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 6273 | . 2 ⊢ (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ↾ ∪ ∪ 𝑅) | |
2 | relresfld 6285 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | |
3 | 1, 2 | eqtrid 2780 | 1 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∪ cuni 4912 I cid 5579 ↾ cres 5684 ∘ ccom 5686 Rel wrel 5687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 |
This theorem is referenced by: relexpsucl 15018 tsrdir 18603 |
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