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Theorem imaundi 4763
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 4652 . . . 4 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
21rneqi 4589 . . 3 ran (𝐴 ↾ (𝐵𝐶)) = ran ((𝐴𝐵) ∪ (𝐴𝐶))
3 rnun 4759 . . 3 ran ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
42, 3eqtri 2076 . 2 ran (𝐴 ↾ (𝐵𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
5 df-ima 4385 . 2 (𝐴 “ (𝐵𝐶)) = ran (𝐴 ↾ (𝐵𝐶))
6 df-ima 4385 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
7 df-ima 4385 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
86, 7uneq12i 3122 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
94, 5, 83eqtr4i 2086 1 (𝐴 “ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cun 2942  ran crn 4373  cres 4374  cima 4375
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 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-xp 4378  df-cnv 4380  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385
This theorem is referenced by:  fnimapr  5260
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