Theorem imass2 5046

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 4974	. . 3 |- ( A C_ B -> ( C |` A ) C_ ( C |` B ) )
2		rnss 4897	. . 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 4677	. 2 |- ( C " A ) = ran ( C |` A )
5		df-ima 4677	. 2 |- ( C " B ) = ran ( C |` B )
6	3, 4, 5	3sstr4g 3227	1 |- ( A C_ B -> ( C " A ) C_ ( C " B ) )

Syntax hints:   -> wi 4   C_ wss 3157   ran crn 4665   |` cres 4666   "cima 4667

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677

This theorem is referenced by:  funimass1  5336  funimass2  5337  fvimacnv  5680  f1imass  5824  ecinxp  6678  sbthlem1  7032  sbthlem2  7033  iscnp4  14538  cnptopco  14542  cnntri  14544  cnrest2  14556  cnptopresti  14558  cnptoprest  14559  metcnp3  14831