Theorem iinab 3991

Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.)

Assertion

Ref	Expression
iinab	⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem iinab

Step	Hyp	Ref	Expression
1		nfcv 2349	. . . 4 ⊢ Ⅎ𝑦𝐴
2		nfab1 2351	. . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑}
3	1, 2	nfiinxy 3956	. . 3 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑}
4		nfab1 2351	. . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}
5	3, 4	cleqf 2374	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}))
6		abid 2194	. . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑)
7	6	ralbii 2513	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
8		vex 2776	. . . 4 ⊢ 𝑦 ∈ V
9		eliin 3934	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}))
10	8, 9	ax-mp 5	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})
11		abid 2194	. . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑)
12	7, 10, 11	3bitr4i 212	. 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑})
13	5, 12	mpgbir 1477	1 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}

Syntax hints:   ↔ wb 105   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  Vcvv 2773  ∩ ciin 3930

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-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-v 2775  df-iin 3932

This theorem is referenced by:  iinrabm  3992