Theorem imass1 5361

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 5289	. . 3 |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
2		rnss 5221	. . 3 |- ( ( A |` C ) C_ ( B |` C ) -> ran ( A |` C ) C_ ran ( B |` C ) )
3	1, 2	syl 16	. 2 |- ( A C_ B -> ran ( A |` C ) C_ ran ( B |` C ) )
4		df-ima 5002	. 2 |- ( A " C ) = ran ( A |` C )
5		df-ima 5002	. 2 |- ( B " C ) = ran ( B |` C )
6	3, 4, 5	3sstr4g 3530	1 |- ( A C_ B -> ( A " C ) C_ ( B " C ) )

Syntax hints:   -> wi 4   C_ wss 3461   ran crn 4990   |` cres 4991   "cima 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002

This theorem is referenced by:  vdwnnlem1  14390  gsumzresOLD  16792  gsumzaddOLD  16811  gsum2dOLD  16874  dprdfaddOLD  16941  dprdres  16949  imasnopn  20064  imasncld  20065  imasncls  20066  tsmsresOLD  20518  utoptop  20610  restutop  20613  ustuqtop3  20619  utopreg  20628  metustblOLD  20956  metustbl  20957  imadifxp  27330  eulerpartlemmf  28187  hess  37484