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Theorem imauni 6141
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 4326 . . 3  |-  U. B  =  U_ x  e.  B  x
21imaeq2i 5157 . 2  |-  ( A
" U. B )  =  ( A " U_ x  e.  B  x )
3 imaiun 6140 . 2  |-  ( A
" U_ x  e.  B  x )  =  U_ x  e.  B  ( A " x )
42, 3eqtri 2433 1  |-  ( A
" U. B )  =  U_ x  e.  B  ( A "
x )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1407   U.cuni 4193   U_ciun 4273   "cima 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-xp 4831  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838
This theorem is referenced by:  enfin2i  8735  tgcn  20048  cncmp  20187  qtoptop2  20494  mbfimaopnlem  22356
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