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Mirrors > Home > ILE Home > Th. List > relcoi2 | 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 4908 | . . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
2 | unss 3324 | . . . . 5 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) ↔ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅) | |
3 | simpr 110 | . . . . 5 ⊢ ((dom 𝑅 ⊆ ∪ ∪ 𝑅 ∧ ran 𝑅 ⊆ ∪ ∪ 𝑅) → ran 𝑅 ⊆ ∪ ∪ 𝑅) | |
4 | 2, 3 | sylbir 135 | . . . 4 ⊢ ((dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 → ran 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ ran 𝑅 ⊆ ∪ ∪ 𝑅 |
6 | cores 5150 | . . 3 ⊢ (ran 𝑅 ⊆ ∪ ∪ 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)) | |
7 | 5, 6 | mp1i 10 | . 2 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = ( I ∘ 𝑅)) |
8 | coi2 5163 | . 2 ⊢ (Rel 𝑅 → ( I ∘ 𝑅) = 𝑅) | |
9 | 7, 8 | eqtrd 2222 | 1 ⊢ (Rel 𝑅 → (( I ↾ ∪ ∪ 𝑅) ∘ 𝑅) = 𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∪ cun 3142 ⊆ wss 3144 ∪ cuni 3824 I cid 4306 dom cdm 4644 ran crn 4645 ↾ cres 4646 ∘ ccom 4648 Rel wrel 4649 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 |
This theorem is referenced by: (None) |
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