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Mirrors > Home > MPE Home > Th. List > res0 | Structured version Visualization version GIF version |
Description: A restriction to the empty set is empty. (Contributed by NM, 12-Nov-1994.) |
Ref | Expression |
---|---|
res0 | ⊢ (𝐴 ↾ ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5261 | . 2 ⊢ (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V)) | |
2 | 0xp 5339 | . . 3 ⊢ (∅ × V) = ∅ | |
3 | 2 | ineq2i 3962 | . 2 ⊢ (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅) |
4 | in0 4112 | . 2 ⊢ (𝐴 ∩ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2797 | 1 ⊢ (𝐴 ↾ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1631 Vcvv 3351 ∩ cin 3722 ∅c0 4063 × cxp 5247 ↾ cres 5251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pr 5034 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-sn 4317 df-pr 4319 df-op 4323 df-opab 4847 df-xp 5255 df-res 5261 |
This theorem is referenced by: ima0 5622 resdisj 5704 smo0 7608 tfrlem16 7642 tz7.44-1 7655 mapunen 8285 fnfi 8394 ackbij2lem3 9265 hashf1lem1 13441 setsid 16121 meet0 17345 join0 17346 frmdplusg 17599 psgn0fv0 18138 gsum2dlem2 18577 ablfac1eulem 18679 ablfac1eu 18680 psrplusg 19596 ply1plusgfvi 19827 ptuncnv 21831 ptcmpfi 21837 ust0 22243 xrge0gsumle 22856 xrge0tsms 22857 jensen 24936 egrsubgr 26392 0grsubgr 26393 pthdlem1 26897 0pth 27305 1pthdlem1 27315 eupth2lemb 27417 resf1o 29845 gsumle 30119 xrge0tsmsd 30125 esumsnf 30466 dfpo2 31983 eldm3 31989 rdgprc0 32035 zrdivrng 34084 eldioph4b 37901 diophren 37903 ismeannd 41201 psmeasure 41205 isomennd 41265 hoidmvlelem3 41331 aacllem 43078 |
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