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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 4357 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | inss1 3157 | . 2 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ 𝐴 | |
3 | 1, 2 | eqsstri 2975 | 1 ⊢ (𝐴 ↾ 𝐵) ⊆ 𝐴 |
Colors of variables: wff set class |
Syntax hints: Vcvv 2557 ∩ cin 2916 ⊆ wss 2917 × cxp 4343 ↾ cres 4347 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-in 2924 df-ss 2931 df-res 4357 |
This theorem is referenced by: relssres 4648 resexg 4650 iss 4654 relresfld 4847 relcoi1 4849 funres 4941 funres11 4971 funcnvres 4972 2elresin 5010 fssres 5066 foimacnv 5144 tposss 5861 dftpos4 5878 smores 5907 smores2 5909 |
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