Theorem imaundir 5141

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 4731	. . 3 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ran ((𝐴 ∪ 𝐵) ↾ 𝐶)
2		resundir 5018	. . . 4 ⊢ ((𝐴 ∪ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))
3	2	rneqi 4951	. . 3 ⊢ ran ((𝐴 ∪ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶))
4		rnun 5136	. . 3 ⊢ ran ((𝐴 ↾ 𝐶) ∪ (𝐵 ↾ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
5	1, 3, 4	3eqtri 2254	. 2 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
6		df-ima 4731	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
7		df-ima 4731	. . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
8	6, 7	uneq12i 3356	. 2 ⊢ ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∪ ran (𝐵 ↾ 𝐶))
9	5, 8	eqtr4i 2253	1 ⊢ ((𝐴 ∪ 𝐵) “ 𝐶) = ((𝐴 “ 𝐶) ∪ (𝐵 “ 𝐶))

Syntax hints:   = wceq 1395   ∪ cun 3195  ran crn 4719   ↾ cres 4720   “ cima 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731

This theorem is referenced by:  fvun1  5699