Theorem imaundi 6003

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 5862	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 5802	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 5999	. . 3 ⊢ ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
4	2, 3	eqtri 2844	. 2 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
5		df-ima 5563	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 5563	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 5563	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
8	6, 7	uneq12i 4137	. 2 ⊢ ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶)) = (ran (𝐴 ↾ 𝐵) ∪ ran (𝐴 ↾ 𝐶))
9	4, 5, 8	3eqtr4i 2854	1 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ((𝐴 “ 𝐵) ∪ (𝐴 “ 𝐶))

Syntax hints:   = wceq 1533   ∪ cun 3934  ran crn 5551   ↾ cres 5552   “ cima 5553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-cnv 5558  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563

This theorem is referenced by:  fnimapr  6742  domunfican  8785  fiint  8789  fodomfi  8791  marypha1lem  8891  resunimafz0  13797  dprd2da  19158  dmdprdsplit2lem  19161  uniioombllem3  24180  mbfimaicc  24226  plyeq0  24795  fnimatp  30417  ffsrn  30459  tocyccntz  30781  noetalem4  33215  imadifss  34861  poimirlem1  34887  poimirlem2  34888  poimirlem3  34889  poimirlem4  34890  poimirlem6  34892  poimirlem7  34893  poimirlem11  34897  poimirlem12  34898  poimirlem15  34901  poimirlem16  34902  poimirlem17  34903  poimirlem19  34905  poimirlem20  34906  poimirlem23  34909  poimirlem24  34910  poimirlem25  34911  poimirlem29  34915  poimirlem31  34917  mbfposadd  34933  itg2addnclem2  34938  ftc1anclem1  34961  ftc1anclem5  34965  brtrclfv2  40065  frege77d  40084  frege109d  40095  frege131d  40102  dffrege76  40278  icccncfext  42162