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Theorem imauni 5761
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) = 𝑥𝐵 (𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem imauni
StepHypRef Expression
1 uniiun 3940 . . 3 𝐵 = 𝑥𝐵 𝑥
21imaeq2i 4968 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
3 imaiun 5760 . 2 (𝐴 𝑥𝐵 𝑥) = 𝑥𝐵 (𝐴𝑥)
42, 3eqtri 2198 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff set class
Syntax hints:   = wceq 1353   cuni 3809   ciun 3886  cima 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by:  tgcn  13678
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