Theorem imaeq1 4948

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 4885	. . 3 |- ( A = B -> ( A |` C ) = ( B |` C ) )
2	1	rneqd 4840	. 2 |- ( A = B -> ran ( A |` C ) = ran ( B |` C ) )
3		df-ima 4624	. 2 |- ( A " C ) = ran ( A |` C )
4		df-ima 4624	. 2 |- ( B " C ) = ran ( B |` C )
5	2, 3, 4	3eqtr4g 2228	1 |- ( A = B -> ( A " C ) = ( B " C ) )

Syntax hints:   -> wi 4   = wceq 1348   ran crn 4612   |` cres 4613   "cima 4614

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  imaeq1i  4950  imaeq1d  4952  eceq2  6550  iscnp  12993