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Theorem iinuni 4545
 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 3064 . . . 4 (∀𝑥𝐵 (𝑦𝐴𝑦𝑥) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
2 elun 3715 . . . . 5 (𝑦 ∈ (𝐴𝑥) ↔ (𝑦𝐴𝑦𝑥))
32ralbii 2963 . . . 4 (∀𝑥𝐵 𝑦 ∈ (𝐴𝑥) ↔ ∀𝑥𝐵 (𝑦𝐴𝑦𝑥))
4 vex 3176 . . . . . 6 𝑦 ∈ V
54elint2 4417 . . . . 5 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
65orbi2i 540 . . . 4 ((𝑦𝐴𝑦 𝐵) ↔ (𝑦𝐴 ∨ ∀𝑥𝐵 𝑦𝑥))
71, 3, 63bitr4ri 292 . . 3 ((𝑦𝐴𝑦 𝐵) ↔ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥))
87abbii 2726 . 2 {𝑦 ∣ (𝑦𝐴𝑦 𝐵)} = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
9 df-un 3545 . 2 (𝐴 𝐵) = {𝑦 ∣ (𝑦𝐴𝑦 𝐵)}
10 df-iin 4458 . 2 𝑥𝐵 (𝐴𝑥) = {𝑦 ∣ ∀𝑥𝐵 𝑦 ∈ (𝐴𝑥)}
118, 9, 103eqtr4i 2642 1 (𝐴 𝐵) = 𝑥𝐵 (𝐴𝑥)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896   ∪ cun 3538  ∩ cint 4410  ∩ ciin 4456 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-v 3175  df-un 3545  df-int 4411  df-iin 4458 This theorem is referenced by: (None)
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