Theorem imaeq1 4941

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

Step	Hyp	Ref	Expression
1		reseq1 4878	. . 3 |- ( A = B -> ( A |` C ) = ( B |` C ) )
2	1	rneqd 4833	. 2 |- ( A = B -> ran ( A |` C ) = ran ( B |` C ) )
3		df-ima 4617	. 2 |- ( A " C ) = ran ( A |` C )
4		df-ima 4617	. 2 |- ( B " C ) = ran ( B |` C )
5	2, 3, 4	3eqtr4g 2224	1 |- ( A = B -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1343   ran crn 4605   |` cres 4606   "cima 4607

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  imaeq1i  4943  imaeq1d  4945  eceq2  6538  iscnp  12839