Theorem rneqi 5008

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 5007	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	ax-mp 5	1 |- ran A = ran B

Syntax hints:   = wceq 1402   ran crn 4773

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-cnv 4780  df-dm 4782  df-rn 4783

This theorem is referenced by:  rnmpt  5028  resima  5094  resima2  5095  mptima  5136  ima0  5144  rnuni  5197  imaundi  5198  imaundir  5199  inimass  5202  dminxp  5230  imainrect  5231  xpima1  5232  xpima2m  5233  rnresv  5245  imacnvcnv  5250  rnpropg  5265  imadmres  5278  mptpreima  5279  dmco  5294  resdif  5659  fpr  5891  fprg  5892  fliftfuns  5997  rnoprab  6164  rnmpo  6192  qliftfuns  6886  xpassen  7121  sbthlemi6  7272  ennnfonelemrn  13291  cnconst2  15260  elply2  15762  iedgedgg  16219  edgiedgbg  16223  edg0iedg0g  16224  uhgrvtxedgiedgb  16301  uspgrf1oedg  16334  usgrf1oedg  16363  usgredg3  16372  ushgredgedg  16384  ushgredgedgloop  16386  0grsubgr  16422  edginwlkd  16513