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Theorem imauni 5325
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 B) = x B (Ax)
Distinct variable groups:   x,A   x,B

Proof of Theorem imauni
StepHypRef Expression
1 uniiun 3684 . . 3 B = x B x
21imaeq2i 4593 . 2 (A B) = (A x B x)
3 imaiun 5324 . 2 (A x B x) = x B (Ax)
42, 3eqtri 2042 1 (A B) = x B (Ax)
Colors of variables: wff set class
Syntax hints:   = wceq 1228   cuni 3554   ciun 3631  cima 4275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285
This theorem is referenced by: (None)
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