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Theorem rneqi 4735
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
StepHypRef Expression
1 rneqi.1 . 2 𝐴 = 𝐵
2 rneq 4734 . 2 (𝐴 = 𝐵 → ran 𝐴 = ran 𝐵)
31, 2ax-mp 5 1 ran 𝐴 = ran 𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1314  ran crn 4508
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-cnv 4515  df-dm 4517  df-rn 4518
This theorem is referenced by:  rnmpt  4755  resima  4820  resima2  4821  ima0  4866  rnuni  4918  imaundi  4919  imaundir  4920  inimass  4923  dminxp  4951  imainrect  4952  xpima1  4953  xpima2m  4954  rnresv  4966  imacnvcnv  4971  rnpropg  4986  imadmres  4999  mptpreima  5000  dmco  5015  resdif  5355  fpr  5568  fprg  5569  fliftfuns  5665  rnoprab  5820  rnmpo  5847  qliftfuns  6479  xpassen  6690  sbthlemi6  6816  ennnfonelemrn  11838  cnconst2  12308
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