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

Theorem iunxun 5094
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 4196 . . . 4 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
2 eliun 4995 . . . . 5 (𝑦 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑦𝐶)
3 eliun 4995 . . . . 5 (𝑦 𝑥𝐵 𝐶 ↔ ∃𝑥𝐵 𝑦𝐶)
42, 3orbi12i 915 . . . 4 ((𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶) ↔ (∃𝑥𝐴 𝑦𝐶 ∨ ∃𝑥𝐵 𝑦𝐶))
51, 4bitr4i 278 . . 3 (∃𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
6 eliun 4995 . . 3 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴𝐵)𝑦𝐶)
7 elun 4153 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶) ↔ (𝑦 𝑥𝐴 𝐶𝑦 𝑥𝐵 𝐶))
85, 6, 73bitr4i 303 . 2 (𝑦 𝑥 ∈ (𝐴𝐵)𝐶𝑦 ∈ ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶))
98eqriv 2734 1 𝑥 ∈ (𝐴𝐵)𝐶 = ( 𝑥𝐴 𝐶 𝑥𝐵 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wo 848   = wceq 1540  wcel 2108  wrex 3070  cun 3949   ciun 4991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3071  df-v 3482  df-un 3956  df-iun 4993
This theorem is referenced by:  iunxdif3  5095  iunxprg  5096  iunsuc  6469  funiunfv  7268  iunfi  9383  kmlem11  10201  ackbij1lem9  10267  fsum2dlem  15806  fsumiun  15857  fprod2dlem  16016  prmreclem4  16957  fiuncmp  23412  ovolfiniun  25536  finiunmbl  25579  volfiniun  25582  voliunlem1  25585  uniioombllem4  25621  iuninc  32573  iunxunsn  32579  iunxunpr  32580  ofpreima2  32676  indval2  32839  esum2dlem  34093  sigaclfu2  34122  fiunelros  34175  measvuni  34215  cvmliftlem10  35299  mrsubvrs  35527  mblfinlem2  37665  dfrcl4  43689  iunrelexp0  43715  comptiunov2i  43719  corclrcl  43720  trclfvdecomr  43741  dfrtrcl4  43751  corcltrcl  43752  cotrclrcl  43755  fiiuncl  45070  iunp1  45071  sge0iunmptlemfi  46428  ovolval4lem1  46664
  Copyright terms: Public domain W3C validator