Theorem rneqi 4987

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 4986	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	ax-mp 5	1 |- ran A = ran B

Syntax hints:   = wceq 1398   ran crn 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-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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-cnv 4759  df-dm 4761  df-rn 4762

This theorem is referenced by:  rnmpt  5007  resima  5073  resima2  5074  mptima  5115  ima0  5123  rnuni  5176  imaundi  5177  imaundir  5178  inimass  5181  dminxp  5209  imainrect  5210  xpima1  5211  xpima2m  5212  rnresv  5224  imacnvcnv  5229  rnpropg  5244  imadmres  5257  mptpreima  5258  dmco  5273  resdif  5638  fpr  5868  fprg  5869  fliftfuns  5973  rnoprab  6138  rnmpo  6166  qliftfuns  6855  xpassen  7083  sbthlemi6  7234  ennnfonelemrn  13187  cnconst2  15115  elply2  15617  iedgedgg  16073  edgiedgbg  16077  edg0iedg0g  16078  uhgrvtxedgiedgb  16155  uspgrf1oedg  16188  usgrf1oedg  16217  usgredg3  16226  ushgredgedg  16238  ushgredgedgloop  16240  0grsubgr  16276  edginwlkd  16367