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Theorem rneq 4662
Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
rneq  |-  ( A  =  B  ->  ran  A  =  ran  B )

Proof of Theorem rneq
StepHypRef Expression
1 cnveq 4610 . . 3  |-  ( A  =  B  ->  `' A  =  `' B
)
21dmeqd 4638 . 2  |-  ( A  =  B  ->  dom  `' A  =  dom  `' B )
3 df-rn 4449 . 2  |-  ran  A  =  dom  `' A
4 df-rn 4449 . 2  |-  ran  B  =  dom  `' B
52, 3, 43eqtr4g 2145 1  |-  ( A  =  B  ->  ran  A  =  ran  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   `'ccnv 4437   dom cdm 4438   ran crn 4439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by:  rneqi  4663  rneqd  4664  xpima1  4877  feq1  5145  foeq1  5229
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