Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  viin GIF version

Theorem viin 3811
 Description: Indexed intersection with a universal index class. (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 3755 . 2 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴}
2 ralv 2650 . . 3 (∀𝑥 ∈ V 𝑦𝐴 ↔ ∀𝑥 𝑦𝐴)
32abbii 2210 . 2 {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴} = {𝑦 ∣ ∀𝑥 𝑦𝐴}
41, 3eqtri 2115 1 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
 Colors of variables: wff set class Syntax hints:  ∀wal 1294   = wceq 1296   ∈ wcel 1445  {cab 2081  ∀wral 2370  Vcvv 2633  ∩ ciin 3753 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-11 1449  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-ral 2375  df-v 2635  df-iin 3755 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator