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Mirrors > Home > ILE Home > Th. List > resss | GIF version |
Description: A class includes its restriction. Exercise 15 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
resss | ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 4599 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | inss1 3327 | . 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 3160 | 1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: Vcvv 2712 ∩ cin 3101 ⊆ wss 3102 × cxp 4585 ↾ cres 4589 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-in 3108 df-ss 3115 df-res 4599 |
This theorem is referenced by: relssres 4905 resexg 4907 iss 4913 cocnvres 5111 relresfld 5116 relcoi1 5118 funres 5212 funres11 5243 funcnvres 5244 2elresin 5282 fssres 5346 foimacnv 5433 tposss 6194 dftpos4 6211 smores 6240 smores2 6242 caserel 7032 txss12 12708 txbasval 12709 |
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