Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  imass1 GIF version

Theorem imass1 4700
 Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 4637 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 4564 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 14 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 4358 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 4358 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 2986 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ⊆ wss 2917  ran crn 4346   ↾ cres 4347   “ cima 4348 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator