Theorem imass2 5112

Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
imass2	|- ( A C_ B -> ( C " A ) C_ ( C " B ) )

Proof of Theorem imass2

Step	Hyp	Ref	Expression
1		ssres2 5040	. . 3 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
2		rnss 4962	. . 3 |- ( ( C |` A ) C_ ( C |` B ) -> ran ( C |` A ) C_ ran ( C |` B ) )
3	1, 2	syl 14	. 2 |- ( A C_ B -> ran ( C |` A ) C_ ran ( C |` B ) )
4		df-ima 4738	. 2 |- ( C " A ) = ran ( C |` A )
5		df-ima 4738	. 2 |- ( C " B ) = ran ( C |` B )
6	3, 4, 5	3sstr4g 3270	1 |- ( A C_ B -> ( C " A ) C_ ( C " B ) )

Syntax hints:   -> wi 4   C_ wss 3200   ran crn 4726   |` cres 4727   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  funimass1  5407  funimass2  5408  fvimacnv  5762  fnfvimad  5889  f1imass  5914  ecinxp  6778  sbthlem1  7155  sbthlem2  7156  iscnp4  14941  cnptopco  14945  cnntri  14947  cnrest2  14959  cnptopresti  14961  cnptoprest  14962  metcnp3  15234