Theorem imass2 5042

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 4970	. . 3 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
2		rnss 4893	. . 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 4673	. 2 |- ( C " A ) = ran ( C |` A )
5		df-ima 4673	. 2 |- ( C " B ) = ran ( C |` B )
6	3, 4, 5	3sstr4g 3223	1 |- ( A C_ B -> ( C " A ) C_ ( C " B ) )

Syntax hints:   -> wi 4   C_ wss 3154   ran crn 4661   |` cres 4662   "cima 4663

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673

This theorem is referenced by:  funimass1  5332  funimass2  5333  fvimacnv  5674  f1imass  5818  ecinxp  6666  sbthlem1  7018  sbthlem2  7019  iscnp4  14397  cnptopco  14401  cnntri  14403  cnrest2  14415  cnptopresti  14417  cnptoprest  14418  metcnp3  14690