Theorem dfrn4 4957

Description: Range defined in terms of image. (Contributed by NM, 14-May-2008.)

Assertion

Ref	Expression
dfrn4	|- ran A = ( A " _V )

Proof of Theorem dfrn4

Step	Hyp	Ref	Expression
1		df-ima 4512	. 2 |- ( A " _V ) = ran ( A |` _V )
2		rnresv 4956	. 2 |- ran ( A |` _V ) = ran A
3	1, 2	eqtr2i 2136	1 |- ran A = ( A " _V )

Syntax hints:   = wceq 1314   _Vcvv 2657   ran crn 4500   |` cres 4501   "cima 4502

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  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 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512

This theorem is referenced by:  dmmpt  4992  ctssdccl  6948