Theorem resopab 5045

Description: Restriction of a class abstraction of ordered pairs. (Contributed by NM, 5-Nov-2002.)

Assertion

Ref	Expression
resopab	⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem resopab

Step	Hyp	Ref	Expression
1		df-res 4728	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V))
2		df-xp 4722	. . . . . 6 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
3		vex 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
4	3	biantru 302	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V))
5	4	opabbii 4150	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ V)}
6	2, 5	eqtr4i 2253	. . . . 5 ⊢ (𝐴 × V) = {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}
7	6	ineq2i 3402	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴})
8		incom 3396	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴}) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
9	7, 8	eqtri 2250	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
10		inopab 4851	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝑥 ∈ 𝐴} ∩ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
11	9, 10	eqtri 2250	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∩ (𝐴 × V)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}
12	1, 11	eqtri 2250	1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}

Syntax hints:   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∩ cin 3196  {copab 4143   × cxp 4714   ↾ cres 4718

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4722  df-rel 4723  df-res 4728

This theorem is referenced by:  resopab2  5048  opabresid  5054  mptpreima  5218  isarep2  5404  resoprab  6091  df1st2  6355  df2nd2  6356