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Theorem iinuni 5100
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 3189 . . . 4 (∀𝑥𝐵 (𝑦𝐴𝑦𝑥) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
2 elun 4147 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
32ralbii 3091 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
4 vex 3476 . . . . . 6 𝑦 ∈ V
54elint2 4956 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
65orbi2i 909 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
71, 3, 63bitr4ri 303 . . 3 ((𝑦𝐴𝑦 𝐵) ↔ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87abbii 2800 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3952 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 4999 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103eqtr4i 2768 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wo 843   = wceq 1539  wcel 2104  {cab 2707  wral 3059  cun 3945   cint 4949   ciin 4997
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-v 3474  df-un 3952  df-int 4950  df-iin 4999
This theorem is referenced by: (None)
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