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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 5872 | . 2 ⊢ (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = (𝑅 ↾ ∪ ∪ 𝑅) | |
2 | relresfld 5881 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ∪ ∪ 𝑅) = 𝑅) | |
3 | 1, 2 | syl5eq 2845 | 1 ⊢ (Rel 𝑅 → (𝑅 ∘ ( I ↾ ∪ ∪ 𝑅)) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∪ cuni 4628 I cid 5219 ↾ cres 5314 ∘ ccom 5316 Rel wrel 5317 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pr 5097 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3387 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-id 5220 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 |
This theorem is referenced by: relexpsucl 14114 tsrdir 17553 |
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