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Mirrors > Home > ILE Home > Th. List > res0 | 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 4659 | . 2 ⊢ (𝐴 ↾ ∅) = (𝐴 ∩ (∅ × V)) | |
2 | 0xp 4727 | . . 3 ⊢ (∅ × V) = ∅ | |
3 | 2 | ineq2i 3348 | . 2 ⊢ (𝐴 ∩ (∅ × V)) = (𝐴 ∩ ∅) |
4 | in0 3472 | . 2 ⊢ (𝐴 ∩ ∅) = ∅ | |
5 | 1, 3, 4 | 3eqtri 2214 | 1 ⊢ (𝐴 ↾ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 Vcvv 2752 ∩ cin 3143 ∅c0 3437 × cxp 4645 ↾ cres 4649 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-opab 4083 df-xp 4653 df-res 4659 |
This theorem is referenced by: ima0 5008 resdisj 5078 smo0 6327 tfr0dm 6351 tfr0 6352 fnfi 6970 setsslid 12574 |
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