Theorem imaundi 4959

Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
imaundi	⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

Proof of Theorem imaundi

Step	Hyp	Ref	Expression
1		resundi 4840	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 4775	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 4955	. . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
4	2, 3	eqtri 2161	. 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
5		df-ima 4560	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 4560	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 4560	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
8	6, 7	uneq12i 3233	. 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
9	4, 5, 8	3eqtr4i 2171	1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

Syntax hints:   = wceq 1332   ∪ cun 3074  ran crn 4548   ↾ cres 4549   “ cima 4550

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by:  fnimapr  5489  fiintim  6825  fidcenumlemrks  6849  fidcenumlemr  6851  resunimafz0  10606  ennnfonelemhf1o  11962