Theorem imauni 5804

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 3966	. . 3 |- U. B = U_ x e. B x
2	1	imaeq2i 5003	. 2 |- ( A " U. B ) = ( A " U_ x e. B x )
3		imaiun 5803	. 2 |- ( A " U_ x e. B x ) = U_ x e. B ( A " x )
4	2, 3	eqtri 2214	1 |- ( A " U. B ) = U_ x e. B ( A " x )

Syntax hints:   = wceq 1364   U.cuni 3835   U_ciun 3912   "cima 4662

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-xp 4665  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by:  tgcn  14376