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Theorem iunxun 3804
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 3178 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
2 eliun 3729 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 3729 . . . . 5 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
42, 3orbi12i 716 . . . 4 ((𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶) ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
51, 4bitr4i 185 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
6 eliun 3729 . . 3 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴𝐵)𝑦𝐶)
7 elun 3139 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶) ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
85, 6, 73bitr4i 210 . 2 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶))
98eqriv 2085 1 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wo 664   = wceq 1289  wcel 1438  wrex 2360  cun 2995   ciun 3725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-iun 3727
This theorem is referenced by:  iunsuc  4238  rdgisuc1  6131  oasuc  6207  omsuc  6215  iunfidisj  6634  fsum2dlemstep  10791  fsumiun  10833
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