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Theorem imauni 7194
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 5019 . . 3 𝐵 = 𝑥𝐵 𝑥
21imaeq2i 6012 . 2 (𝐴 𝐵) = (𝐴 𝑥𝐵 𝑥)
3 imaiun 7193 . 2 (𝐴 𝑥𝐵 𝑥) = 𝑥𝐵 (𝐴𝑥)
42, 3eqtri 2761 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   cuni 4866   ciun 4955  cima 5637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647
This theorem is referenced by:  enfin2i  10262  tgcn  22619  cncmp  22759  qtoptop2  23066  mbfimaopnlem  25035  fnpreimac  31633
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