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Theorem imaeq1 4846
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )

Proof of Theorem imaeq1
StepHypRef Expression
1 reseq1 4783 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 4738 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4522 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4522 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2175 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1316   ran crn 4510    |` cres 4511   "cima 4512
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  imaeq1i  4848  imaeq1d  4850  eceq2  6434  iscnp  12295
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