Theorem imauni 5811

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 3971	. . 3 |- U. B = U_ x e. B x
2	1	imaeq2i 5008	. 2 |- ( A " U. B ) = ( A " U_ x e. B x )
3		imaiun 5810	. 2 |- ( A " U_ x e. B x ) = U_ x e. B ( A " x )
4	2, 3	eqtri 2217	1 |- ( A " U. B ) = U_ x e. B ( A " x )

Syntax hints:   = wceq 1364   U.cuni 3840   U_ciun 3917   "cima 4667

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-iun 3919  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677

This theorem is referenced by:  tgcn  14528