Theorem imass1 5500

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 5424	. . 3 |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
2		rnss 5354	. . 3 |- ( ( A |` C ) C_ ( B |` C ) -> ran ( A |` C ) C_ ran ( B |` C ) )
3	1, 2	syl 17	. 2 |- ( A C_ B -> ran ( A |` C ) C_ ran ( B |` C ) )
4		df-ima 5127	. 2 |- ( A " C ) = ran ( A |` C )
5		df-ima 5127	. 2 |- ( B " C ) = ran ( B |` C )
6	3, 4, 5	3sstr4g 3646	1 |- ( A C_ B -> ( A " C ) C_ ( B " C ) )

Syntax hints:   -> wi 4   C_ wss 3574   ran crn 5115   |` cres 5116   "cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  vdwnnlem1  15699  dprdres  18427  imasnopn  21493  imasncld  21494  imasncls  21495  utoptop  22038  restutop  22041  ustuqtop3  22047  utopreg  22056  metustbl  22371  imadifxp  29414  esum2d  30155  eulerpartlemmf  30437  brtrclfv2  38019  frege97d  38044  frege109d  38049  frege131d  38056  hess  38074  resimass  39449