Theorem opres 2580

Description: Ordered pair membership in a restriction when the first member belongs to the restricting class.

Hypothesis

Ref	Expression
opres.1	⊢ B ∈ V

Assertion

Ref	Expression
opres	⊢ (A ∈ D → (⟨A, B⟩ ∈ (C ↾ D) ↔ ⟨A, B⟩ ∈ C))

Proof of Theorem opres

Step	Hyp	Ref	Expression
1		opres.1	. . . . 5 ⊢ B ∈ V
2	1	opelres 2579	. . . 4 ⊢ (⟨A, B⟩ ∈ (C ↾ D) ↔ (⟨A, B⟩ ∈ C ∧ A ∈ D))
3	2	pm3.26bd 259	. . 3 ⊢ (⟨A, B⟩ ∈ (C ↾ D) → ⟨A, B⟩ ∈ C)
4	3	a1i 7	. 2 ⊢ (A ∈ D → (⟨A, B⟩ ∈ (C ↾ D) → ⟨A, B⟩ ∈ C))
5	2	biimpr 134	. . . 4 ⊢ ((⟨A, B⟩ ∈ C ∧ A ∈ D) → ⟨A, B⟩ ∈ (C ↾ D))
6	5	exp 291	. . 3 ⊢ (⟨A, B⟩ ∈ C → (A ∈ D → ⟨A, B⟩ ∈ (C ↾ D)))
7	6	com12 13	. 2 ⊢ (A ∈ D → (⟨A, B⟩ ∈ C → ⟨A, B⟩ ∈ (C ↾ D)))
8	4, 7	impbid 397	1 ⊢ (A ∈ D → (⟨A, B⟩ ∈ (C ↾ D) ↔ ⟨A, B⟩ ∈ C))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   ↾ cres 2412

This theorem is referenced by:  resieq 2581  2elresin 2733

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-res 2430

metamath.org