Theorem imass1 4807

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 4739	. . 3 |- ( A C_ B -> ( A |` C ) C_ ( B |` C ) )
2		rnss 4665	. . 3 |- ( ( A |` C ) C_ ( B |` C ) -> ran ( A |` C ) C_ ran ( B |` C ) )
3	1, 2	syl 14	. 2 |- ( A C_ B -> ran ( A |` C ) C_ ran ( B |` C ) )
4		df-ima 4451	. 2 |- ( A " C ) = ran ( A |` C )
5		df-ima 4451	. 2 |- ( B " C ) = ran ( B |` C )
6	3, 4, 5	3sstr4g 3067	1 |- ( A C_ B -> ( A " C ) C_ ( B " C ) )

Syntax hints:   -> wi 4   C_ wss 2999   ran crn 4439   |` cres 4440   "cima 4441

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451

This theorem is referenced by: (None)