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Mirrors > Home > ILE Home > Th. List > ima0 | GIF version |
Description: Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.) |
Ref | Expression |
---|---|
ima0 | ⊢ (𝐴 “ ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ima 4358 | . 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅) | |
2 | res0 4616 | . . 3 ⊢ (𝐴 ↾ ∅) = ∅ | |
3 | 2 | rneqi 4562 | . 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅ |
4 | rn0 4588 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2064 | 1 ⊢ (𝐴 “ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∅c0 3224 ran crn 4346 ↾ cres 4347 “ cima 4348 |
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-in1 544 ax-in2 545 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-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 |
This theorem is referenced by: (None) |
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