Theorem rneqi 4984

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 4983	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	ax-mp 5	1 |- ran A = ran B

Syntax hints:   = wceq 1398   ran crn 4749

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 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-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-cnv 4756  df-dm 4758  df-rn 4759

This theorem is referenced by:  rnmpt  5004  resima  5070  resima2  5071  mptima  5112  ima0  5120  rnuni  5173  imaundi  5174  imaundir  5175  inimass  5178  dminxp  5206  imainrect  5207  xpima1  5208  xpima2m  5209  rnresv  5221  imacnvcnv  5226  rnpropg  5241  imadmres  5254  mptpreima  5255  dmco  5270  resdif  5635  fpr  5865  fprg  5866  fliftfuns  5970  rnoprab  6135  rnmpo  6163  qliftfuns  6852  xpassen  7080  sbthlemi6  7231  ennnfonelemrn  13159  cnconst2  15085  elply2  15587  iedgedgg  16043  edgiedgbg  16047  edg0iedg0g  16048  uhgrvtxedgiedgb  16125  uspgrf1oedg  16158  usgrf1oedg  16187  usgredg3  16196  ushgredgedg  16208  ushgredgedgloop  16210  0grsubgr  16246  edginwlkd  16337