Theorem imass1 5015

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 4945	. . 3 |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
2		rnss 4869	. . 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 4651	. 2 |- ( A " C ) = ran ( A |` C )
5		df-ima 4651	. 2 |- ( B " C ) = ran ( B |` C )
6	3, 4, 5	3sstr4g 3210	1 |- ( A C_ B -> ( A " C ) C_ ( B " C ) )

Syntax hints:   -> wi 4   C_ wss 3141   ran crn 4639   |` cres 4640   "cima 4641

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651

This theorem is referenced by:  imasnopn  14070