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Theorem imaundi 4946
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 4827 . . . 4 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
21rneqi 4762 . . 3 ran (𝐴 ↾ (𝐵𝐶)) = ran ((𝐴𝐵) ∪ (𝐴𝐶))
3 rnun 4942 . . 3 ran ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
42, 3eqtri 2158 . 2 ran (𝐴 ↾ (𝐵𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
5 df-ima 4547 . 2 (𝐴 “ (𝐵𝐶)) = ran (𝐴 ↾ (𝐵𝐶))
6 df-ima 4547 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
7 df-ima 4547 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
86, 7uneq12i 3223 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
94, 5, 83eqtr4i 2168 1 (𝐴 “ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1331  cun 3064  ran crn 4535  cres 4536  cima 4537
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547
This theorem is referenced by:  fnimapr  5474  fiintim  6810  fidcenumlemrks  6834  fidcenumlemr  6836  resunimafz0  10567  ennnfonelemhf1o  11915
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