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Theorem ima0 4778
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 4441 . 2 (𝐴 “ ∅) = ran (𝐴 ↾ ∅)
2 res0 4705 . . 3 (𝐴 ↾ ∅) = ∅
32rneqi 4651 . 2 ran (𝐴 ↾ ∅) = ran ∅
4 rn0 4677 . 2 ran ∅ = ∅
51, 3, 43eqtri 2112 1 (𝐴 “ ∅) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1289  c0 3284  ran crn 4429  cres 4430  cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by:  fidcenumlemrk  6642  fidcenumlemr  6643
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