Theorem imaeq1 5069

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 5005	. . 3 |- ( A = B -> ( A |` C ) = ( B |` C ) )
2	1	rneqd 4959	. 2 |- ( A = B -> ran ( A |` C ) = ran ( B |` C ) )
3		df-ima 4736	. 2 |- ( A " C ) = ran ( A |` C )
4		df-ima 4736	. 2 |- ( B " C ) = ran ( B |` C )
5	2, 3, 4	3eqtr4g 2287	1 |- ( A = B -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1395   ran crn 4724   |` cres 4725   "cima 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  imaeq1i  5071  imaeq1d  5073  eceq2  6734  iscnp  14913  elply2  15449