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Theorem imaundi 4831
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 4714 . . . 4 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
21rneqi 4651 . . 3 ran (𝐴 ↾ (𝐵𝐶)) = ran ((𝐴𝐵) ∪ (𝐴𝐶))
3 rnun 4827 . . 3 ran ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
42, 3eqtri 2108 . 2 ran (𝐴 ↾ (𝐵𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
5 df-ima 4441 . 2 (𝐴 “ (𝐵𝐶)) = ran (𝐴 ↾ (𝐵𝐶))
6 df-ima 4441 . . 3 (𝐴𝐵) = ran (𝐴𝐵)
7 df-ima 4441 . . 3 (𝐴𝐶) = ran (𝐴𝐶)
86, 7uneq12i 3150 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = (ran (𝐴𝐵) ∪ ran (𝐴𝐶))
94, 5, 83eqtr4i 2118 1 (𝐴 “ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  cun 2995  ran crn 4429  cres 4430  cima 4431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-cnv 4436  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441
This theorem is referenced by:  fnimapr  5348  fidcenumlemrks  6641  fidcenumlemr  6643  resunimafz0  10201
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