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Theorem iunin1 4719
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4707 to recover Enderton's theorem. (Contributed by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
iunin1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin1
StepHypRef Expression
1 iunin2 4718 . 2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
2 incom 3956 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 11 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iuneq2i 4673 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 3956 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4i 2803 1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  cin 3722   ciun 4654
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
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-iun 4656
This theorem is referenced by:  2iunin  4722  resiun1  5555  tgrest  21180  metnrmlem3  22880  limciun  23874  uniin1  29701  disjunsn  29741  measinblem  30619  sstotbnd2  33901  subsaliuncl  41090  sge0iunmptlemre  41146
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