Theorem imaundi 6149

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 5994	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 5929	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 6144	. . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
4	2, 3	eqtri 2788	. 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
5		df-ima 5676	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 5676	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 5676	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
8	6, 7	uneq12i 4120	. 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
9	4, 5, 8	3eqtr4i 2798	1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 3904  ran crn 5664   ↾ cres 5665   “ cima 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  cnvimassrndm  6151  fnimapr  6968  fnimatpd  6969  naddasslem1  8683  naddasslem2  8684  fodomfi  9275  domunfican  9284  fiint  9289  marypha1lem  9396  resunimafz0  14495  dprd2da  20137  dmdprdsplit2lem  20140  uniioombllem3  25773  mbfimaicc  25819  plyeq0  26397  madeoldsuc  28107  addbday  28240  negbdaylem  28278  bdaypw2n0bndlem  28685  ffsrn  33102  tocyccntz  33487  imadifss  38279  poimirlem1  38305  poimirlem2  38306  poimirlem3  38307  poimirlem4  38308  poimirlem6  38310  poimirlem7  38311  poimirlem11  38315  poimirlem12  38316  poimirlem15  38319  poimirlem16  38320  poimirlem17  38321  poimirlem19  38323  poimirlem20  38324  poimirlem23  38327  poimirlem24  38328  poimirlem25  38329  poimirlem29  38333  poimirlem31  38335  mbfposadd  38351  itg2addnclem2  38356  ftc1anclem1  38377  ftc1anclem5  38381  brtrclfv2  44486  frege77d  44505  frege109d  44516  frege131d  44523  dffrege76  44698  icccncfext  46634  cycl3grtri  48745