Theorem imaundi 6130

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 5975	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 5909	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 6125	. . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
4	2, 3	eqtri 2784	. 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
5		df-ima 5656	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 5656	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 5656	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
8	6, 7	uneq12i 4117	. 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
9	4, 5, 8	3eqtr4i 2794	1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

Syntax hints:   = wceq 1559   ∪ cun 3900  ran crn 5644   ↾ cres 5645   “ cima 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-xp 5649  df-cnv 5651  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656

This theorem is referenced by:  cnvimassrndm  6133  fnimapr  6945  fnimatpd  6946  naddasslem1  8659  naddasslem2  8660  fodomfi  9250  imafiOLD  9254  domunfican  9260  fiint  9265  marypha1lem  9373  resunimafz0  14452  dprd2da  20075  dmdprdsplit2lem  20078  uniioombllem3  25635  mbfimaicc  25681  plyeq0  26259  madeoldsuc  27966  addbday  28099  negbdaylem  28137  bdaypw2n0bndlem  28544  ffsrn  32891  tocyccntz  33285  imadifss  38055  poimirlem1  38081  poimirlem2  38082  poimirlem3  38083  poimirlem4  38084  poimirlem6  38086  poimirlem7  38087  poimirlem11  38091  poimirlem12  38092  poimirlem15  38095  poimirlem16  38096  poimirlem17  38097  poimirlem19  38099  poimirlem20  38100  poimirlem23  38103  poimirlem24  38104  poimirlem25  38105  poimirlem29  38109  poimirlem31  38111  mbfposadd  38127  itg2addnclem2  38132  ftc1anclem1  38153  ftc1anclem5  38157  brtrclfv2  44264  frege77d  44283  frege109d  44294  frege131d  44301  dffrege76  44476  icccncfext  46422  cycl3grtri  48530