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Mirrors > Home > ILE Home > Th. List > coires1 | GIF version |
Description: Composition with a restricted identity relation. (Contributed by FL, 19-Jun-2011.) (Revised by Stefan O'Rear, 7-Mar-2015.) |
Ref | Expression |
---|---|
coires1 | ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cocnvcnv1 5057 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I ) | |
2 | relcnv 4925 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
3 | coi1 5062 | . . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴 |
5 | 1, 4 | eqtr3i 2163 | . . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴 |
6 | 5 | reseq1i 4823 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) |
7 | resco 5051 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) | |
8 | 6, 7 | eqtr3i 2163 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) |
9 | rescnvcnv 5009 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
10 | 8, 9 | eqtr3i 2163 | 1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 I cid 4218 ◡ccnv 4546 ↾ cres 4549 ∘ ccom 4551 Rel wrel 4552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 |
This theorem is referenced by: funcoeqres 5406 |
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