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Theorem imauni 5400
 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 3710 . . 3 𝐵 = 𝑥𝐵 𝑥
21imaeq2i 4666 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
3 imaiun 5399 . 2 (𝐴 𝑥𝐵 𝑥) = 𝑥𝐵 (𝐴𝑥)
42, 3eqtri 2060 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ∪ cuni 3580  ∪ ciun 3657   “ cima 4348 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358 This theorem is referenced by: (None)
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