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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 5160 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = (𝐴 ∘ I ) | |
2 | relcnv 5027 | . . . . . 6 ⊢ Rel ◡◡𝐴 | |
3 | coi1 5165 | . . . . . 6 ⊢ (Rel ◡◡𝐴 → (◡◡𝐴 ∘ I ) = ◡◡𝐴) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (◡◡𝐴 ∘ I ) = ◡◡𝐴 |
5 | 1, 4 | eqtr3i 2212 | . . . 4 ⊢ (𝐴 ∘ I ) = ◡◡𝐴 |
6 | 5 | reseq1i 4924 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (◡◡𝐴 ↾ 𝐵) |
7 | resco 5154 | . . 3 ⊢ ((𝐴 ∘ I ) ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) | |
8 | 6, 7 | eqtr3i 2212 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ∘ ( I ↾ 𝐵)) |
9 | rescnvcnv 5112 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
10 | 8, 9 | eqtr3i 2212 | 1 ⊢ (𝐴 ∘ ( I ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 I cid 4309 ◡ccnv 4646 ↾ cres 4649 ∘ ccom 4651 Rel wrel 4652 |
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 4139 ax-pow 4195 ax-pr 4230 |
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 3595 df-sn 3616 df-pr 3617 df-op 3619 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 |
This theorem is referenced by: funcoeqres 5514 |
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