Theorem viin 3986

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 3929	. 2 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴}
2		ralv 2788	. . 3 ⊢ (∀𝑥 ∈ V 𝑦 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴)
3	2	abbii 2320	. 2 ⊢ {𝑦 ∣ ∀𝑥 ∈ V 𝑦 ∈ 𝐴} = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}
4	1, 3	eqtri 2225	1 ⊢ ∩ 𝑥 ∈ V 𝐴 = {𝑦 ∣ ∀𝑥 𝑦 ∈ 𝐴}

Syntax hints:  ∀wal 1370   = wceq 1372   ∈ wcel 2175  {cab 2190  ∀wral 2483  Vcvv 2771  ∩ ciin 3927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-ral 2488  df-v 2773  df-iin 3929

This theorem is referenced by: (None)