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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 4686 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
| 2 | inss1 3392 | . 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴 | |
| 3 | 1, 2 | eqsstri 3224 | 1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 |
| Colors of variables: wff set class |
| Syntax hints: Vcvv 2771 ∩ cin 3164 ⊆ wss 3165 × cxp 4672 ↾ cres 4676 |
| 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 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-ext 2186 |
| This theorem depends on definitions: df-bi 117 df-tru 1375 df-nf 1483 df-sb 1785 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-v 2773 df-in 3171 df-ss 3178 df-res 4686 |
| This theorem is referenced by: relssres 4996 resexg 4998 iss 5004 cocnvres 5206 relresfld 5211 relcoi1 5213 funres 5311 funres11 5345 funcnvres 5346 2elresin 5386 fssres 5450 foimacnv 5539 tposss 6331 dftpos4 6348 smores 6377 smores2 6379 caserel 7188 txss12 14709 txbasval 14710 |
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