Theorem imaeq2 4877

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 4814	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4768	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4552	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4552	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> ( C " A ) = ( C " B ) )

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-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552

This theorem is referenced by:  imaeq2i  4879  imaeq2d  4881  ssimaex  5482  ssimaexg  5483  isoselem  5721  f1opw2  5976  fopwdom  6730  ssenen  6745  fiintim  6817  fidcenumlemrk  6842  fidcenumlemr  6843  sbthlem2  6846  isbth  6855  ennnfonelemp1  11919  ennnfonelemnn0  11935  ctinfomlemom  11940  ctinfom  11941  tgcn  12377  iscnp4  12387  cnpnei  12388  cnima  12389  cnconst2  12402  cnrest2  12405  cnptoprest  12408  txcnp  12440  txcnmpt  12442  metcnp3  12680