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Theorem viin 4956
 Description: Indexed intersection with a universal index class. When 𝐴 doesn't depend on 𝑥, this evaluates to 𝐴 by 19.3 2200 and abid2 2894. When 𝐴 = 𝑥, this evaluates to ∅ by intiin 4951 and intv 5236. (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 4889 . 2 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴}
2 ralv 3435 . . 3 (∀𝑥 ∈ V 𝑦𝐴 ↔ ∀𝑥 𝑦𝐴)
32abbii 2823 . 2 {𝑦 ∣ ∀𝑥 ∈ V 𝑦𝐴} = {𝑦 ∣ ∀𝑥 𝑦𝐴}
41, 3eqtri 2781 1 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦𝐴}
 Colors of variables: wff setvar class Syntax hints:  ∀wal 1536   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409  ∩ ciin 4887 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-iin 4889 This theorem is referenced by: (None)
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