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

Theorem viin 4506
Description: Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2055 and abid2 2728. When 𝐴 = 𝑥, this evaluates to by intiin 4501 and intv 4759. (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 4449 . 2 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴}
2 ralv 3188 . . 3 (∀𝑥 ∈ V 𝑦𝐴 ↔ ∀𝑥 𝑦𝐴)
32abbii 2722 . 2 {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴} = {𝑦 ∣ ∀𝑥 𝑦𝐴}
41, 3eqtri 2628 1 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
Colors of variables: wff setvar class
Syntax hints:  wal 1472   = wceq 1474  wcel 1976  {cab 2592  wral 2892  Vcvv 3169   ciin 4447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-ral 2897  df-v 3171  df-iin 4449
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator