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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 5560 | . 2 ⊢ (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V)) | |
2 | 0xp 5642 | . . 3 ⊢ (∅ × V) = ∅ | |
3 | 2 | ineq2i 4183 | . 2 ⊢ (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅) |
4 | in0 4342 | . 2 ⊢ (𝐴 ∩ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2845 | 1 ⊢ (𝐴 ↾ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Vcvv 3492 ∩ cin 3932 ∅c0 4288 × cxp 5546 ↾ cres 5550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 df-res 5560 |
This theorem is referenced by: ima0 5938 resdisj 6019 smo0 7984 tfrlem16 8018 tz7.44-1 8031 mapunen 8674 fnfi 8784 ackbij2lem3 9651 hashf1lem1 13801 setsid 16526 meet0 17735 join0 17736 frmdplusg 18007 psgn0fv0 18568 gsum2dlem2 19020 ablfac1eulem 19123 ablfac1eu 19124 psrplusg 20089 ply1plusgfvi 20338 ptuncnv 22343 ptcmpfi 22349 ust0 22755 xrge0gsumle 23368 xrge0tsms 23369 jensen 25493 egrsubgr 26986 0grsubgr 26987 pthdlem1 27474 0pth 27831 1pthdlem1 27841 eupth2lemb 27943 resf1o 30392 xrge0tsmsd 30619 gsumle 30652 esumsnf 31222 satfv1lem 32506 dfpo2 32888 eldm3 32894 rdgprc0 32935 zrdivrng 35112 eldioph4b 39286 diophren 39288 ismeannd 42626 psmeasure 42630 isomennd 42690 hoidmvlelem3 42756 aacllem 44830 |
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