Theorem imaundi 6115

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 5960	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 5894	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 6111	. . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
4	2, 3	eqtri 2760	. 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
5		df-ima 5645	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 5645	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 5645	. . 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 2770	1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

Syntax hints:   = wceq 1542   ∪ cun 3901  ran crn 5633   ↾ cres 5634   “ cima 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645

This theorem is referenced by:  cnvimassrndm  6118  fnimapr  6925  fnimatpd  6926  naddasslem1  8632  naddasslem2  8633  fodomfi  9224  imafiOLD  9228  domunfican  9234  fiint  9239  fodomfiOLD  9242  marypha1lem  9348  resunimafz0  14380  dprd2da  19985  dmdprdsplit2lem  19988  uniioombllem3  25554  mbfimaicc  25600  plyeq0  26184  madeoldsuc  27893  addbday  28026  negbdaylem  28064  bdaypw2n0bndlem  28471  ffsrn  32817  tocyccntz  33237  imadifss  37843  poimirlem1  37869  poimirlem2  37870  poimirlem3  37871  poimirlem4  37872  poimirlem6  37874  poimirlem7  37875  poimirlem11  37879  poimirlem12  37880  poimirlem15  37883  poimirlem16  37884  poimirlem17  37885  poimirlem19  37887  poimirlem20  37888  poimirlem23  37891  poimirlem24  37892  poimirlem25  37893  poimirlem29  37897  poimirlem31  37899  mbfposadd  37915  itg2addnclem2  37920  ftc1anclem1  37941  ftc1anclem5  37945  brtrclfv2  44080  frege77d  44099  frege109d  44110  frege131d  44117  dffrege76  44292  icccncfext  46242  cycl3grtri  48304