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Theorem rneq 4813
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 4760 . . 3  |-  ( A  =  B  ->  `' A  =  `' B
)
21dmeqd 4788 . 2  |-  ( A  =  B  ->  dom  `' A  =  dom  `' B )
3 df-rn 4597 . 2  |-  ran  A  =  dom  `' A
4 df-rn 4597 . 2  |-  ran  B  =  dom  `' B
52, 3, 43eqtr4g 2215 1  |-  ( A  =  B  ->  ran  A  =  ran  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335   `'ccnv 4585   dom cdm 4586   ran crn 4587
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597
This theorem is referenced by:  rneqi  4814  rneqd  4815  xpima1  5032  feq1  5302  foeq1  5388  ixpsnf1o  6681
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