Theorem imass1 4922

Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)

Assertion

Ref	Expression
imass1	|- ( A C_ B -> ( A " C ) C_ ( B " C ) )

Proof of Theorem imass1

Step	Hyp	Ref	Expression
1		ssres 4853	. . 3 |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
2		rnss 4777	. . 3 |- ( ( A |` C ) C_ ( B |` C ) -> ran ( A |` C ) C_ ran ( B |` C ) )
3	1, 2	syl 14	. 2 |- ( A C_ B -> ran ( A |` C ) C_ ran ( B |` C ) )
4		df-ima 4560	. 2 |- ( A " C ) = ran ( A |` C )
5		df-ima 4560	. 2 |- ( B " C ) = ran ( B |` C )
6	3, 4, 5	3sstr4g 3145	1 |- ( A C_ B -> ( A " C ) C_ ( B " C ) )

Syntax hints:   -> wi 4   C_ wss 3076   ran crn 4548   |` cres 4549   "cima 4550

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by:  imasnopn  12507