Theorem imass2 4980

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 4911	. . 3 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
2		rnss 4834	. . 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 4617	. 2 |- ( C " A ) = ran ( C |` A )
5		df-ima 4617	. 2 |- ( C " B ) = ran ( C |` B )
6	3, 4, 5	3sstr4g 3185	1 |- ( A C_ B -> ( C " A ) C_ ( C " B ) )

Syntax hints:   -> wi 4   C_ wss 3116   ran crn 4605   |` cres 4606   "cima 4607

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  funimass1  5265  funimass2  5266  fvimacnv  5600  f1imass  5742  ecinxp  6576  sbthlem1  6922  sbthlem2  6923  iscnp4  12858  cnptopco  12862  cnntri  12864  cnrest2  12876  cnptopresti  12878  cnptoprest  12879  metcnp3  13151