Theorem imass2 5067

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 4995	. . 3 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
2		rnss 4917	. . 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 4696	. 2 |- ( C " A ) = ran ( C |` A )
5		df-ima 4696	. 2 |- ( C " B ) = ran ( C |` B )
6	3, 4, 5	3sstr4g 3240	1 |- ( A C_ B -> ( C " A ) C_ ( C " B ) )

Syntax hints:   -> wi 4   C_ wss 3170   ran crn 4684   |` cres 4685   "cima 4686

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-br 4052  df-opab 4114  df-xp 4689  df-cnv 4691  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696

This theorem is referenced by:  funimass1  5360  funimass2  5361  fvimacnv  5708  f1imass  5856  ecinxp  6710  sbthlem1  7074  sbthlem2  7075  iscnp4  14765  cnptopco  14769  cnntri  14771  cnrest2  14783  cnptopresti  14785  cnptoprest  14786  metcnp3  15058