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Theorem imauni 5624
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 3853 . . 3  |-  U. B  =  U_ x  e.  B  x
21imaeq2i 4917 . 2  |-  ( A
" U. B )  =  ( A " U_ x  e.  B  x )
3 imaiun 5623 . 2  |-  ( A
" U_ x  e.  B  x )  =  U_ x  e.  B  ( A " x )
42, 3eqtri 2273 1  |-  ( A
" U. B )  =  U_ x  e.  B  ( A "
x )
Colors of variables: wff set class
Syntax hints:    = wceq 1619   U.cuni 3727   U_ciun 3803   "cima 4583
This theorem is referenced by:  enfin2i  7831  tgcn  16814  cncmp  16951  qtoptop2  17222  mbfimaopnlem  18842
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601
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