Theorem imauni 5729

Description: The image of a union is the indexed union of the images. Theorem 3K(a) of [Enderton] p. 50. (Contributed by NM, 9-Aug-2004.) (Proof shortened by Mario Carneiro, 18-Jun-2014.)

Assertion

Ref	Expression
imauni	|- ( A " U. B ) = U_ x e. B ( A " x )

Distinct variable groups:   x, A   x, B

Proof of Theorem imauni

Step	Hyp	Ref	Expression
1		uniiun 3919	. . 3 |- U. B = U_ x e. B x
2	1	imaeq2i 4944	. 2 |- ( A " U. B ) = ( A " U_ x e. B x )
3		imaiun 5728	. 2 |- ( A " U_ x e. B x ) = U_ x e. B ( A " x )
4	2, 3	eqtri 2186	1 |- ( A " U. B ) = U_ x e. B ( A " x )

Syntax hints:   = wceq 1343   U.cuni 3789   U_ciun 3866   "cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  tgcn  12848