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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 5305 | . . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
2 | unss 3749 | . . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) | |
3 | simpr 476 | . . . 4 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) → ran 𝑅 ⊆ ∪ ∪ 𝑅) | |
4 | 2, 3 | sylbir 224 | . . 3 ⊢ ((dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | cores 5555 | . . 3 ⊢ (ran 𝑅 ⊆ ∪ ∪ 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)) | |
6 | 1, 4, 5 | mp2b 10 | . 2 ⊢ (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅) |
7 | coi2 5569 | . 2 ⊢ (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅) | |
8 | 6, 7 | syl5eq 2656 | 1 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∪ cun 3538 ⊆ wss 3540 ∪ cuni 4372 I cid 4948 dom cdm 5038 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 |
This theorem is referenced by: relexpsucr 13617 tsrdir 17061 |
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