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

Theorem iuniin 4893
Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iuniin 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuniin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 r19.12 3234 . . . 4 (∃𝑥𝐴𝑦𝐵 𝑧𝐶 → ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
2 eliin 4886 . . . . . 6 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶))
32elv 3404 . . . . 5 (𝑧 𝑦𝐵 𝐶 ↔ ∀𝑦𝐵 𝑧𝐶)
43rexbii 3161 . . . 4 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴𝑦𝐵 𝑧𝐶)
5 eliun 4885 . . . . 5 (𝑧 𝑥𝐴 𝐶 ↔ ∃𝑥𝐴 𝑧𝐶)
65ralbii 3080 . . . 4 (∀𝑦𝐵 𝑧 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵𝑥𝐴 𝑧𝐶)
71, 4, 63imtr4i 295 . . 3 (∃𝑥𝐴 𝑧 𝑦𝐵 𝐶 → ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
8 eliun 4885 . . 3 (𝑧 𝑥𝐴 𝑦𝐵 𝐶 ↔ ∃𝑥𝐴 𝑧 𝑦𝐵 𝐶)
9 eliin 4886 . . . 4 (𝑧 ∈ V → (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶))
109elv 3404 . . 3 (𝑧 𝑦𝐵 𝑥𝐴 𝐶 ↔ ∀𝑦𝐵 𝑧 𝑥𝐴 𝐶)
117, 8, 103imtr4i 295 . 2 (𝑧 𝑥𝐴 𝑦𝐵 𝐶𝑧 𝑦𝐵 𝑥𝐴 𝐶)
1211ssriv 3881 1 𝑥𝐴 𝑦𝐵 𝐶 𝑦𝐵 𝑥𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wb 209  wcel 2114  wral 3053  wrex 3054  Vcvv 3398  wss 3843   ciun 4881   ciin 4882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rex 3059  df-v 3400  df-in 3850  df-ss 3860  df-iun 4883  df-iin 4884
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator