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Theorem iunxun 5098
Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iunxun 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)

Proof of Theorem iunxun
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 rexun 4188 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
2 eliun 5001 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 5001 . . . . 5 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
42, 3orbi12i 912 . . . 4 ((𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶) ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
51, 4bitr4i 277 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
6 eliun 5001 . . 3 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴𝐵)𝑦𝐶)
7 elun 4145 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶) ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
85, 6, 73bitr4i 302 . 2 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶))
98eqriv 2722 1 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wo 845   = wceq 1533  wcel 2098  wrex 3059  cun 3942   ciun 4997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rex 3060  df-v 3463  df-un 3949  df-iun 4999
This theorem is referenced by:  iunxdif3  5099  iunxprg  5100  iunsuc  6456  funiunfv  7258  iunfi  9371  kmlem11  10190  ackbij1lem9  10258  fsum2dlem  15760  fsumiun  15811  fprod2dlem  15968  prmreclem4  16907  fiuncmp  23369  ovolfiniun  25491  finiunmbl  25534  volfiniun  25537  voliunlem1  25540  uniioombllem4  25576  iuninc  32450  iunxunsn  32456  iunxunpr  32457  ofpreima2  32553  indval2  33784  esum2dlem  33862  sigaclfu2  33891  fiunelros  33944  measvuni  33984  cvmliftlem10  35055  mrsubvrs  35283  mblfinlem2  37282  dfrcl4  43253  iunrelexp0  43279  comptiunov2i  43283  corclrcl  43284  trclfvdecomr  43305  dfrtrcl4  43315  corcltrcl  43316  cotrclrcl  43319  fiiuncl  44576  iunp1  44577  sge0iunmptlemfi  45944  ovolval4lem1  46180
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