MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  iunin2 Structured version   Visualization version   GIF version

Theorem iunin2 4898
Description: Indexed union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use uniiun 4887 to recover Enderton's theorem. (Contributed by NM, 26-Mar-2004.)
Assertion
Ref Expression
iunin2 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iunin2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.42v 3313 . . . 4 (∃𝑥𝐴 (𝑦𝐵𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝑦𝐶))
2 elin 4096 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
32rexbii 3213 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵𝑦𝐶))
4 eliun 4835 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
54anbi2i 622 . . . 4 ((𝑦𝐵𝑦 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 𝑦𝐶))
61, 3, 53bitr4i 304 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
7 eliun 4835 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
8 elin 4096 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵𝑦 𝑥𝐴 𝐶))
96, 7, 83bitr4i 304 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
109eqriv 2794 1 𝑥𝐴 (𝐵𝐶) = (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1525  wcel 2083  wrex 3108  cin 3864   ciun 4831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-v 3442  df-in 3872  df-iun 4833
This theorem is referenced by:  iunin1  4899  2iunin  4903  resiun2  5762  infssuni  8668  kmlem11  9439  cmpsublem  21695  cmpsub  21696  kgentopon  21834  metnrmlem3  23156  ovoliunlem1  23790  voliunlem1  23838  voliunlem2  23839  uniioombllem2  23871  uniioombllem4  23874  volsup2  23893  itg1addlem5  23988  itg1climres  24002  uniin2  29989  carsgclctunlem2  31190  cvmscld  32130  cnambfre  34492  ftc1anclem6  34524  heiborlem3  34644  carageniuncllem2  42368
  Copyright terms: Public domain W3C validator