Theorem imauni 5740

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 3926	. . 3 |- U. B = U_ x e. B x
2	1	imaeq2i 4951	. 2 |- ( A " U. B ) = ( A " U_ x e. B x )
3		imaiun 5739	. 2 |- ( A " U_ x e. B x ) = U_ x e. B ( A " x )
4	2, 3	eqtri 2191	1 |- ( A " U. B ) = U_ x e. B ( A " x )

Syntax hints:   = wceq 1348   U.cuni 3796   U_ciun 3873   "cima 4614

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by:  tgcn  13002