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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 4522 | . 2 ⊢ (𝐴 “ ∅) = ran (𝐴 ↾ ∅) | |
2 | res0 4793 | . . 3 ⊢ (𝐴 ↾ ∅) = ∅ | |
3 | 2 | rneqi 4737 | . 2 ⊢ ran (𝐴 ↾ ∅) = ran ∅ |
4 | rn0 4765 | . 2 ⊢ ran ∅ = ∅ | |
5 | 1, 3, 4 | 3eqtri 2142 | 1 ⊢ (𝐴 “ ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1316 ∅c0 3333 ran crn 4510 ↾ cres 4511 “ cima 4512 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 |
This theorem is referenced by: fiintim 6785 fidcenumlemrk 6810 fidcenumlemr 6811 ennnfonelem1 11847 ennnfonelemhf1o 11853 |
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