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Theorem viin 4714
Description: Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2224 and abid2 2894. When 𝐴 = 𝑥, this evaluates to by intiin 4709 and intv 4973. (Contributed by NM, 11-Sep-2008.)
Assertion
Ref Expression
viin 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem viin
StepHypRef Expression
1 df-iin 4658 . 2 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴}
2 ralv 3371 . . 3 (∀𝑥 ∈ V 𝑦𝐴 ↔ ∀𝑥 𝑦𝐴)
32abbii 2888 . 2 {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴} = {𝑦 ∣ ∀𝑥 𝑦𝐴}
41, 3eqtri 2793 1 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
Colors of variables: wff setvar class
Syntax hints:  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  Vcvv 3351   ciin 4656
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-ral 3066  df-v 3353  df-iin 4658
This theorem is referenced by: (None)
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