Theorem ima0 5121

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

Step	Hyp	Ref	Expression
1		df-ima 4762	. 2 |- ( A " (/) ) = ran ( A |` (/) )
2		res0 5042	. . 3 |- ( A |` (/) ) = (/)
3	2	rneqi 4985	. 2 |- ran ( A |` (/) ) = ran (/)
4		rn0 5013	. 2 |- ran (/) = (/)
5	1, 3, 4	3eqtri 2257	1 |- ( A " (/) ) = (/)

Syntax hints:   = wceq 1398   (/)c0 3508   ran crn 4750   |` cres 4751   "cima 4752

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-cnv 4757  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762

This theorem is referenced by:  supp0cosupp0fn  6467  fiintim  7191  fidcenumlemrk  7224  fidcenumlemr  7225  ennnfonelem1  13158  ennnfonelemhf1o  13164  eupth2lembfi  16472