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Theorem imauni 5985
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
StepHypRef Expression
1 uniiun 4136 . . 3  |-  U. B  =  U_ x  e.  B  x
21imaeq2i 5193 . 2  |-  ( A
" U. B )  =  ( A " U_ x  e.  B  x )
3 imaiun 5984 . 2  |-  ( A
" U_ x  e.  B  x )  =  U_ x  e.  B  ( A " x )
42, 3eqtri 2455 1  |-  ( A
" U. B )  =  U_ x  e.  B  ( A "
x )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   U.cuni 4007   U_ciun 4085   "cima 4873
This theorem is referenced by:  enfin2i  8193  tgcn  17308  cncmp  17447  qtoptop2  17723  mbfimaopnlem  19539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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