Theorem rneqi 4907

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 4906	. 2 |- ( A = B -> ran A = ran B )
3	1, 2	ax-mp 5	1 |- ran A = ran B

Syntax hints:   = wceq 1373   ran crn 4677

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-cnv 4684  df-dm 4686  df-rn 4687

This theorem is referenced by:  rnmpt  4927  resima  4993  resima2  4994  mptima  5035  ima0  5042  rnuni  5095  imaundi  5096  imaundir  5097  inimass  5100  dminxp  5128  imainrect  5129  xpima1  5130  xpima2m  5131  rnresv  5143  imacnvcnv  5148  rnpropg  5163  imadmres  5176  mptpreima  5177  dmco  5192  resdif  5546  fpr  5768  fprg  5769  fliftfuns  5869  rnoprab  6030  rnmpo  6058  qliftfuns  6708  xpassen  6927  sbthlemi6  7066  ennnfonelemrn  12823  cnconst2  14738  elply2  15240  iedgedgg  15688  edgiedgbg  15692  edg0iedg0g  15693