Theorem imaundir 6101

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 5631	. . 3 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ran ((𝐴 ∪ 𝐵) ↾ 𝐶)
2		resundir 5946	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))
3	2	rneqi 5879	. . 3 ⊢ ran ((𝐴 ∪ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))
4		rnun 6096	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
5	1, 3, 4	3eqtri 2766	. 2 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
6		df-ima 5631	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
7		df-ima 5631	. . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
8	6, 7	uneq12i 4096	. 2 ⊢ ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
9	5, 8	eqtr4i 2765	1 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶))

Syntax hints:   = wceq 1547   ∪ cun 3881  ran crn 5619   ↾ cres 5620   “ cima 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631

This theorem is referenced by:  fvun  6917  suppun  8124  fsuppun  9290  fpwwe2lem12  10556  ustuqtop1  24224  mbfres2  25630  imadifxp  32690  suppun2  32776  eulerpartlemt  34555  bj-projun  37347  bj-funun  37612  poimirlem3  37990  poimirlem15  38002  brtrclfv2  44171  frege131d  44208  unhe1  44229  frege110  44417  frege133  44440  aacllem  50291