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Theorem imauni 6650
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 4708 . . 3 𝐵 = 𝑥𝐵 𝑥
21imaeq2i 5604 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
3 imaiun 6649 . 2 (𝐴 𝑥𝐵 𝑥) = 𝑥𝐵 (𝐴𝑥)
42, 3eqtri 2793 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631   cuni 4575   ciun 4655  cima 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263
This theorem is referenced by:  enfin2i  9349  tgcn  21277  cncmp  21416  qtoptop2  21723  mbfimaopnlem  23642
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