Theorem viin 3737

Description: Indexed intersection with a universal index class. (Contributed by NM, 11-Sep-2008.)

Assertion

Ref	Expression
viin	⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem viin

Step	Hyp	Ref	Expression
1		df-iin 3681	. 2 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴}
2		ralv 2616	. . 3 ⊢ (∀𝑥 ∈ V 𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴)
3	2	abbii 2194	. 2 ⊢ {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
4	1, 3	eqtri 2101	1 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}

Syntax hints:  ∀wal 1282   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  Vcvv 2601  ∩ ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-v 2603  df-iin 3681

This theorem is referenced by: (None)