Theorem resopab 2598

Description: Restriction of a class abstraction of ordered pairs.

Assertion

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

Distinct variable group(s):   x,y,A

Proof of Theorem resopab

Step	Hyp	Ref	Expression
1		df-res 2430	. 2 ⊢ ({⟨x, y⟩∣φ} ↾ A) = ({⟨x, y⟩∣φ} ∩ (A × V))
2		df-xp 2424	. . . . 5 ⊢ (A × V) = {⟨x, y⟩∣(x ∈ A ∧ y ∈ V)}
3		visset 1350	. . . . . . 7 ⊢ y ∈ V
4	3	biantru 543	. . . . . 6 ⊢ (x ∈ A ↔ (x ∈ A ∧ y ∈ V))
5	4	biopabi 2103	. . . . 5 ⊢ {⟨x, y⟩∣x ∈ A} = {⟨x, y⟩∣(x ∈ A ∧ y ∈ V)}
6	2, 5	eqtr4 1122	. . . 4 ⊢ (A × V) = {⟨x, y⟩∣x ∈ A}
7	6	ineq2i 1642	. . 3 ⊢ ({⟨x, y⟩∣φ} ∩ (A × V)) = ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣x ∈ A})
8		incom 1636	. . 3 ⊢ ({⟨x, y⟩∣φ} ∩ {⟨x, y⟩∣x ∈ A}) = ({⟨x, y⟩∣x ∈ A} ∩ {⟨x, y⟩∣φ})
9	7, 8	eqtr 1119	. 2 ⊢ ({⟨x, y⟩∣φ} ∩ (A × V)) = ({⟨x, y⟩∣x ∈ A} ∩ {⟨x, y⟩∣φ})
10		inopab 2495	. 2 ⊢ ({⟨x, y⟩∣x ∈ A} ∩ {⟨x, y⟩∣φ}) = {⟨x, y⟩∣(x ∈ A ∧ φ)}
11	1, 9, 10	3eqtr 1123	1 ⊢ ({⟨x, y⟩∣φ} ↾ A) = {⟨x, y⟩∣(x ∈ A ∧ φ)}

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∩ cin 1486  {copab 2055   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  f1stres 3096

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-res 2430

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