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Theorem imauni 5952
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 4104 . . 3  |-  U. B  =  U_ x  e.  B  x
21imaeq2i 5160 . 2  |-  ( A
" U. B )  =  ( A " U_ x  e.  B  x )
3 imaiun 5951 . 2  |-  ( A
" U_ x  e.  B  x )  =  U_ x  e.  B  ( A " x )
42, 3eqtri 2424 1  |-  ( A
" U. B )  =  U_ x  e.  B  ( A "
x )
Colors of variables: wff set class
Syntax hints:    = wceq 1649   U.cuni 3975   U_ciun 4053   "cima 4840
This theorem is referenced by:  enfin2i  8157  tgcn  17270  cncmp  17409  qtoptop2  17684  mbfimaopnlem  19500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850
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