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Theorem iunin1 4788
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4776 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 4787 . 2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
2 incom 4015 . . . 4 (𝐶𝐵) = (𝐵𝐶)
32a1i 11 . . 3 (𝑥𝐴 → (𝐶𝐵) = (𝐵𝐶))
43iuneq2i 4742 . 2 𝑥𝐴 (𝐶𝐵) = 𝑥𝐴 (𝐵𝐶)
5 incom 4015 . 2 ( 𝑥𝐴 𝐶𝐵) = (𝐵 𝑥𝐴 𝐶)
61, 4, 53eqtr4i 2849 1 𝑥𝐴 (𝐶𝐵) = ( 𝑥𝐴 𝐶𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1637  wcel 2157  cin 3779   ciun 4723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-in 3787  df-ss 3794  df-iun 4725
This theorem is referenced by:  2iunin  4791  resiun1  5634  tgrest  21198  metnrmlem3  22898  limciun  23895  uniin1  29715  disjunsn  29755  measinblem  30631  sstotbnd2  33903  subsaliuncl  41073  sge0iunmptlemre  41129
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