Theorem rneqi 4775

Description: Equality inference for range. (Contributed by NM, 4-Mar-2004.)

Hypothesis

Ref	Expression
rneqi.1	|- A = B

Assertion

Ref	Expression
rneqi	|- ran A = ran B

Proof of Theorem rneqi

Step	Hyp	Ref	Expression
1		rneqi.1	. 2 |- A = B
2		rneq 4774	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	ax-mp 5	1 |- ran A = ran B

Syntax hints:   = wceq 1332   ran crn 4548

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by:  rnmpt  4795  resima  4860  resima2  4861  ima0  4906  rnuni  4958  imaundi  4959  imaundir  4960  inimass  4963  dminxp  4991  imainrect  4992  xpima1  4993  xpima2m  4994  rnresv  5006  imacnvcnv  5011  rnpropg  5026  imadmres  5039  mptpreima  5040  dmco  5055  resdif  5397  fpr  5610  fprg  5611  fliftfuns  5707  rnoprab  5862  rnmpo  5889  qliftfuns  6521  xpassen  6732  sbthlemi6  6858  ennnfonelemrn  11968  cnconst2  12441