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Theorem imaeq1 4832
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 4769 . . 3  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
21rneqd 4726 . 2  |-  ( A  =  B  ->  ran  ( A  |`  C )  =  ran  ( B  |`  C ) )
3 df-ima 4510 . 2  |-  ( A
" C )  =  ran  ( A  |`  C )
4 df-ima 4510 . 2  |-  ( B
" C )  =  ran  ( B  |`  C )
52, 3, 43eqtr4g 2170 1  |-  ( A  =  B  ->  ( A " C )  =  ( B " C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1312   ran crn 4498    |` cres 4499   "cima 4500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-sn 3497  df-pr 3498  df-op 3500  df-br 3894  df-opab 3948  df-cnv 4505  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510
This theorem is referenced by:  imaeq1i  4834  imaeq1d  4836  eceq2  6418  iscnp  12204
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