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  6888  sbthlem1  7279  sbthlem2  7280  iscnp4  15444  cnptopco  15448  cnntri  15450  cnrest2  15462  cnptopresti  15464  cnptoprest  15465  metcnp3  15737