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

Theorem iinuni 4992
 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 3324 . . . 4 (∀𝑥𝐵 (𝑦𝐴𝑦𝑥) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
2 elun 4100 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
32ralbii 3152 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
4 vex 3473 . . . . . 6 𝑦 ∈ V
54elint2 4855 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
65orbi2i 909 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
71, 3, 63bitr4ri 306 . . 3 ((𝑦𝐴𝑦 𝐵) ↔ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87abbii 2885 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3914 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 4894 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103eqtr4i 2853 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 843   = wceq 1537   ∈ wcel 2114  {cab 2798  ∀wral 3125   ∪ cun 3907  ∩ cint 4848  ∩ ciin 4892 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-v 3472  df-un 3914  df-int 4849  df-iin 4894 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator