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

Theorem iinuni 5027
Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)
Assertion
Ref Expression
iinuni (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinuni
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 r19.32v 3270 . . . 4 (∀𝑥𝐵 (𝑦𝐴𝑦𝑥) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
2 elun 4083 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
32ralbii 3092 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
4 vex 3436 . . . . . 6 𝑦 ∈ V
54elint2 4886 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
65orbi2i 910 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
71, 3, 63bitr4ri 304 . . 3 ((𝑦𝐴𝑦 𝐵) ↔ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87abbii 2808 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3892 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 4927 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103eqtr4i 2776 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wo 844   = wceq 1539  wcel 2106  {cab 2715  wral 3064  cun 3885   cint 4879   ciin 4925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-un 3892  df-int 4880  df-iin 4927
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator