Theorem imaundi 6145

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 5990	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 5925	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 6140	. . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
4	2, 3	eqtri 2792	. 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
5		df-ima 5672	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 5672	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 5672	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
8	6, 7	uneq12i 4128	. 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
9	4, 5, 8	3eqtr4i 2802	1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

Syntax hints:   = wceq 1567   ∪ cun 3911  ran crn 5660   ↾ cres 5661   “ cima 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is referenced by:  cnvimassrndm  6148  fnimapr  6962  fnimatpd  6963  naddasslem1  8677  naddasslem2  8678  fodomfi  9268  domunfican  9277  fiint  9282  marypha1lem  9389  resunimafz0  14478  dprd2da  20110  dmdprdsplit2lem  20113  uniioombllem3  25709  mbfimaicc  25755  plyeq0  26333  madeoldsuc  28040  addbday  28173  negbdaylem  28211  bdaypw2n0bndlem  28618  ffsrn  33010  tocyccntz  33401  imadifss  38129  poimirlem1  38155  poimirlem2  38156  poimirlem3  38157  poimirlem4  38158  poimirlem6  38160  poimirlem7  38161  poimirlem11  38165  poimirlem12  38166  poimirlem15  38169  poimirlem16  38170  poimirlem17  38171  poimirlem19  38173  poimirlem20  38174  poimirlem23  38177  poimirlem24  38178  poimirlem25  38179  poimirlem29  38183  poimirlem31  38185  mbfposadd  38201  itg2addnclem2  38206  ftc1anclem1  38227  ftc1anclem5  38231  brtrclfv2  44338  frege77d  44357  frege109d  44368  frege131d  44375  dffrege76  44550  icccncfext  46486  cycl3grtri  48594