ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imass2 GIF version

Theorem imass2 4885
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 4816 . . 3 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
2 rnss 4739 . . 3 ((𝐶𝐴) ⊆ (𝐶𝐵) → ran (𝐶𝐴) ⊆ ran (𝐶𝐵))
31, 2syl 14 . 2 (𝐴𝐵 → ran (𝐶𝐴) ⊆ ran (𝐶𝐵))
4 df-ima 4522 . 2 (𝐶𝐴) = ran (𝐶𝐴)
5 df-ima 4522 . 2 (𝐶𝐵) = ran (𝐶𝐵)
63, 4, 53sstr4g 3110 1 (𝐴𝐵 → (𝐶𝐴) ⊆ (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wss 3041  ran crn 4510  cres 4511  cima 4512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522
This theorem is referenced by:  funimass1  5170  funimass2  5171  fvimacnv  5503  f1imass  5643  ecinxp  6472  sbthlem1  6813  sbthlem2  6814  iscnp4  12314  cnptopco  12318  cnntri  12320  cnrest2  12332  cnptopresti  12334  cnptoprest  12335  metcnp3  12607
  Copyright terms: Public domain W3C validator