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

Theorem imaundi 4736
 Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
imaundi (𝐴 “ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem imaundi
StepHypRef Expression
1 resundi 4625 . . . 4 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
21rneqi 4562 . . 3 ran (𝐴 ↾ (𝐵𝐶)) = ran ((𝐴𝐵) ∪ (𝐴𝐶))
3 rnun 4732 . . 3 ran ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
42, 3eqtri 2060 . 2 ran (𝐴 ↾ (𝐵𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
5 df-ima 4358 . 2 (𝐴 “ (𝐵𝐶)) = ran (𝐴 ↾ (𝐵𝐶))
6 df-ima 4358 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
7 df-ima 4358 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
86, 7uneq12i 3095 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
94, 5, 83eqtr4i 2070 1 (𝐴 “ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∪ cun 2915  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-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by:  fnimapr  5233
 Copyright terms: Public domain W3C validator