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Theorem ima0 4866
Description: Image of the empty set. Theorem 3.16(ii) of [Monk1] p. 38. (Contributed by NM, 20-May-1998.)
Assertion
Ref Expression
ima0 (𝐴 “ ∅) = ∅

Proof of Theorem ima0
StepHypRef Expression
1 df-ima 4520 . 2 (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2 res0 4791 . . 3 (𝐴 ↾ ∅) = ∅
32rneqi 4735 . 2 ran (𝐴 ↾ ∅) = ran ∅
4 rn0 4763 . 2 ran ∅ = ∅
51, 3, 43eqtri 2140 1 (𝐴 “ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1314  c0 3331  ran crn 4508  cres 4509  cima 4510
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  fiintim  6783  fidcenumlemrk  6808  fidcenumlemr  6809  ennnfonelem1  11826  ennnfonelemhf1o  11832
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