Theorem imaundir 6108

Description: The image of a union. (Contributed by Jeff Hoffman, 17-Feb-2008.)

Assertion

Ref	Expression
imaundir	⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶))

Proof of Theorem imaundir

Step	Hyp	Ref	Expression
1		df-ima 5637	. . 3 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ran ((𝐴 ∪ 𝐵) ↾ 𝐶)
2		resundir 5953	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))
3	2	rneqi 5886	. . 3 ⊢ ran ((𝐴 ∪ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))
4		rnun 6103	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
5	1, 3, 4	3eqtri 2763	. 2 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
6		df-ima 5637	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
7		df-ima 5637	. . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
8	6, 7	uneq12i 4118	. 2 ⊢ ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
9	5, 8	eqtr4i 2762	1 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶))

Syntax hints:   = wceq 1541   ∪ cun 3899  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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637

This theorem is referenced by:  fvun  6924  suppun  8126  fsuppun  9290  fpwwe2lem12  10553  ustuqtop1  24185  mbfres2  25602  imadifxp  32676  suppun2  32763  eulerpartlemt  34528  bj-projun  37195  bj-funun  37457  poimirlem3  37824  poimirlem15  37836  brtrclfv2  43968  frege131d  44005  unhe1  44026  frege110  44214  frege133  44237  aacllem  50046