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Mirrors > Home > MPE Home > Th. List > relcoi2 | Structured version Visualization version GIF version |
Description: Composition with the identity relation restricted to a relation's field. (Contributed by FL, 2-May-2011.) |
Ref | Expression |
---|---|
relcoi2 | ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmrnssfld 5962 | . . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
2 | unss 4179 | . . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) | |
3 | simpr 484 | . . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) → ran 𝑅 ⊆ ∪ ∪ 𝑅) | |
4 | 2, 3 | sylbir 234 | . . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | cores 6241 | . . 3 ⊢ (ran 𝑅 ⊆ ∪ ∪ 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)) | |
6 | 1, 4, 5 | mp2b 10 | . 2 ⊢ (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅) |
7 | coi2 6255 | . 2 ⊢ (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅) | |
8 | 6, 7 | eqtrid 2778 | 1 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∪ cun 3941 ⊆ wss 3943 ∪ cuni 4902 I cid 5566 dom cdm 5669 ran crn 5670 ↾ cres 5671 ∘ ccom 5673 Rel wrel 5674 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 |
This theorem is referenced by: relexpsucr 14983 tsrdir 18567 |
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