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Theorem imass2 5004
Description: Subset theorem for image. Exercise 22(a) of [Enderton] p. 53. (Contributed by NM, 22-Mar-1998.)
Assertion
Ref Expression
imass2 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))

Proof of Theorem imass2
StepHypRef Expression
1 ssres2 4934 . . 3 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
2 rnss 4857 . . 3 ((𝐶𝐴) ⊆ (𝐶𝐵) → ran (𝐶𝐴) ⊆ ran (𝐶𝐵))
31, 2syl 14 . 2 (𝐴𝐵 → ran (𝐶𝐴) ⊆ ran (𝐶𝐵))
4 df-ima 4639 . 2 (𝐶𝐴) = ran (𝐶𝐴)
5 df-ima 4639 . 2 (𝐶𝐵) = ran (𝐶𝐵)
63, 4, 53sstr4g 3198 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3129  ran crn 4627  cres 4628  cima 4629
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by:  funimass1  5293  funimass2  5294  fvimacnv  5631  f1imass  5774  ecinxp  6609  sbthlem1  6955  sbthlem2  6956  iscnp4  13611  cnptopco  13615  cnntri  13617  cnrest2  13629  cnptopresti  13631  cnptoprest  13632  metcnp3  13904
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