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Theorem iunxun 3981
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 3330 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
2 eliun 3905 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 3905 . . . . 5 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
42, 3orbi12i 765 . . . 4 ((𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶) ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
51, 4bitr4i 187 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
6 eliun 3905 . . 3 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴𝐵)𝑦𝐶)
7 elun 3291 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶) ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
85, 6, 73bitr4i 212 . 2 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶))
98eqriv 2186 1 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wo 709   = wceq 1364  wcel 2160  wrex 2469  cun 3142   ciun 3901
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-iun 3903
This theorem is referenced by:  iunxprg  3982  iunsuc  4438  rdgisuc1  6409  oasuc  6489  omsuc  6497  iunfidisj  6975  fsum2dlemstep  11474  fsumiun  11517  fprod2dlemstep  11662  iuncld  14075
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