Theorem imaeq1 5121

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 5057	. . 3 |- ( A = B -> ( A |` C ) = ( B |` C ) )
2	1	rneqd 5011	. 2 |- ( A = B -> ran ( A |` C ) = ran ( B |` C ) )
3		df-ima 4787	. 2 |- ( A " C ) = ran ( A |` C )
4		df-ima 4787	. 2 |- ( B " C ) = ran ( B |` C )
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ( A " C ) = ( B " C ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   ran crn 4775   |` cres 4776   "cima 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787

This theorem is used by:  imaeq1i  5123  imaeq1d  5125  suppval  6477  eceq2  6844  iscnp  15300  elply2  15836