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

Theorem imaundir 4765
Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)
Assertion
Ref Expression
imaundir ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem imaundir
StepHypRef Expression
1 df-ima 4386 . . 3 ((𝐴𝐵) “ 𝐶) = ran ((𝐴𝐵) ↾ 𝐶)
2 resundir 4654 . . . 4 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
32rneqi 4590 . . 3 ran ((𝐴𝐵) ↾ 𝐶) = ran ((𝐴𝐶) ∪ (𝐵𝐶))
4 rnun 4760 . . 3 ran ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
51, 3, 43eqtri 2080 . 2 ((𝐴𝐵) “ 𝐶) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
6 df-ima 4386 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
7 df-ima 4386 . . 3 (𝐵𝐶) = ran (𝐵𝐶)
86, 7uneq12i 3123 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = (ran (𝐴𝐶) ∪ ran (𝐵𝐶))
95, 8eqtr4i 2079 1 ((𝐴𝐵) “ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cun 2943  ran crn 4374  cres 4375  cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by:  fvun1  5267
  Copyright terms: Public domain W3C validator