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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 5561 | . 2 ⊢ (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V)) | |
2 | 0xp 5643 | . . 3 ⊢ (∅ × V) = ∅ | |
3 | 2 | ineq2i 4185 | . 2 ⊢ (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅) |
4 | in0 4344 | . 2 ⊢ (𝐴 ∩ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2848 | 1 ⊢ (𝐴 ↾ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3494 ∩ cin 3934 ∅c0 4290 × cxp 5547 ↾ cres 5551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 df-xp 5555 df-res 5561 |
This theorem is referenced by: ima0 5939 resdisj 6020 smo0 7989 tfrlem16 8023 tz7.44-1 8036 mapunen 8680 fnfi 8790 ackbij2lem3 9657 hashf1lem1 13807 setsid 16532 meet0 17741 join0 17742 frmdplusg 18013 psgn0fv0 18633 gsum2dlem2 19085 ablfac1eulem 19188 ablfac1eu 19189 psrplusg 20155 ply1plusgfvi 20404 ptuncnv 22409 ptcmpfi 22415 ust0 22822 xrge0gsumle 23435 xrge0tsms 23436 jensen 25560 egrsubgr 27053 0grsubgr 27054 pthdlem1 27541 0pth 27898 1pthdlem1 27908 eupth2lemb 28010 resf1o 30460 xrge0tsmsd 30687 gsumle 30720 esumsnf 31318 satfv1lem 32604 dfpo2 32986 eldm3 32992 rdgprc0 33033 zrdivrng 35225 eldioph4b 39401 diophren 39403 ismeannd 42743 psmeasure 42747 isomennd 42807 hoidmvlelem3 42873 aacllem 44896 |
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