Theorem imass2 5166

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 5092	. . 3 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
2		rnss 5014	. . 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 4789	. 2 |- ( C " A ) = ran ( C |` A )
5		df-ima 4789	. 2 |- ( C " B ) = ran ( C |` B )
6	3, 4, 5	3sstr4g 3291	1 |- ( A C_ B -> ( C " A ) C_ ( C " B ) )

This proof depends on syntax axioms:   -> wi 4   C_ wss 3220   ran crn 4777   |` cres 4778   "cima 4779

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4132  df-opab 4194  df-xp 4782  df-cnv 4784  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789

This theorem is used by:  funimass1  5461  funimass2  5462  fvimacnv  5827  fnfvimad  5957  f1imass  5983  ecinxp  6889  sbthlem1  7281  sbthlem2  7282  mplsubgfilemcl  15232  iscnp4  15461  cnptopco  15465  cnntri  15467  cnrest2  15479  cnptopresti  15481  cnptoprest  15482  metcnp3  15754