Theorem imaundi 6107

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 5952	. . . 4 ⊢ (𝐴 ↾ (𝐵 ∪ 𝐶)) = ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
2	1	rneqi 5886	. . 3 ⊢ ran (𝐴 ↾ (𝐵 ∪ 𝐶)) = ran ((𝐴 ↾ 𝐵) ∪ (𝐴 ↾ 𝐶))
3		rnun 6103	. . 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 5637	. 2 ⊢ (𝐴 “ (𝐵 ∪ 𝐶)) = ran (𝐴 ↾ (𝐵 ∪ 𝐶))
6		df-ima 5637	. . 3 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
7		df-ima 5637	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
8	6, 7	uneq12i 4107	. 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 3888  ran crn 5625   ↾ cres 5626   “ cima 5627

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  cnvimassrndm  6110  fnimapr  6917  fnimatpd  6918  naddasslem1  8623  naddasslem2  8624  fodomfi  9215  imafiOLD  9219  domunfican  9225  fiint  9230  fodomfiOLD  9233  marypha1lem  9339  resunimafz0  14398  dprd2da  20010  dmdprdsplit2lem  20013  uniioombllem3  25562  mbfimaicc  25608  plyeq0  26186  madeoldsuc  27891  addbday  28024  negbdaylem  28062  bdaypw2n0bndlem  28469  ffsrn  32816  tocyccntz  33220  imadifss  37930  poimirlem1  37956  poimirlem2  37957  poimirlem3  37958  poimirlem4  37959  poimirlem6  37961  poimirlem7  37962  poimirlem11  37966  poimirlem12  37967  poimirlem15  37970  poimirlem16  37971  poimirlem17  37972  poimirlem19  37974  poimirlem20  37975  poimirlem23  37978  poimirlem24  37979  poimirlem25  37980  poimirlem29  37984  poimirlem31  37986  mbfposadd  38002  itg2addnclem2  38007  ftc1anclem1  38028  ftc1anclem5  38032  brtrclfv2  44172  frege77d  44191  frege109d  44202  frege131d  44209  dffrege76  44384  icccncfext  46333  cycl3grtri  48435