Theorem imaeq1 4876

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 4813	. . 3 |- ( A = B -> ( A |` C ) = ( B |` C ) )
2	1	rneqd 4768	. 2 |- ( A = B -> ran ( A |` C ) = ran ( B |` C ) )
3		df-ima 4552	. 2 |- ( A " C ) = ran ( A |` C )
4		df-ima 4552	. 2 |- ( B " C ) = ran ( B |` C )
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1331   ran crn 4540   |` cres 4541   "cima 4542

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  imaeq1i  4878  imaeq1d  4880  eceq2  6466  iscnp  12368