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Mirrors > Home > ILE Home > Th. List > residm | GIF version |
Description: Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.) |
Ref | Expression |
---|---|
residm | ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3195 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | resabs2 4963 | . 2 ⊢ (𝐵 ⊆ 𝐵 → ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ⊆ wss 3149 ↾ cres 4653 |
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 4143 ax-pow 4199 ax-pr 4234 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2758 df-un 3153 df-in 3155 df-ss 3162 df-pw 3599 df-sn 3620 df-pr 3621 df-op 3623 df-opab 4087 df-xp 4657 df-rel 4658 df-res 4663 |
This theorem is referenced by: resima 4965 fvsnun2 5744 |
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