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Theorem imauni 5772
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 3955 . . 3  |-  U. B  =  U_ x  e.  B  x
21imaeq2i 5010 . 2  |-  ( A
" U. B )  =  ( A " U_ x  e.  B  x )
3 imaiun 5771 . 2  |-  ( A
" U_ x  e.  B  x )  =  U_ x  e.  B  ( A " x )
42, 3eqtri 2303 1  |-  ( A
" U. B )  =  U_ x  e.  B  ( A "
x )
Colors of variables: wff set class
Syntax hints:    = wceq 1623   U.cuni 3827   U_ciun 3905   "cima 4692
This theorem is referenced by:  enfin2i  7947  tgcn  16982  cncmp  17119  qtoptop2  17390  mbfimaopnlem  19010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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