Theorem opelres 5322

Description: Ordered pair membership in a restriction. Exercise 13 of [TakeutiZaring] p. 25. (Contributed by NM, 13-Nov-1995.)

Hypothesis

Ref	Expression
opelres.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opelres	⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Proof of Theorem opelres

Step	Hyp	Ref	Expression
1		df-res 5050	. . 3 ⊢ (𝐶 ↾ 𝐷) = (𝐶 ∩ (𝐷 × V))
2	1	eleq2i 2680	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)))
3		elin 3758	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)))
4		opelres.1	. . . 4 ⊢ 𝐵 ∈ V
5		opelxp 5070	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ V))
6	4, 5	mpbiran2 956	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ 𝐴 ∈ 𝐷)
7	6	anbi2i 726	. 2 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))
8	2, 3, 7	3bitri 285	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Syntax hints:   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  ⟨cop 4131   × cxp 5036   ↾ cres 5040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-res 5050

This theorem is referenced by:  brres  5323  opelresg  5324  opres  5326  dmres  5339  elres  5355  relssres  5357  iss  5367  restidsing  5377  restidsingOLD  5378  asymref  5431  ssrnres  5491  cnvresima  5541  ressn  5588  funssres  5844  fcnvres  5995  fvn0ssdmfun  6258  resiexg  6994  relexpindlem  13651  dprd2dlem1  18263  dprd2da  18264  hausdiag  21258  hauseqlcld  21259  ovoliunlem1  23077  h2hlm  27221  undmrnresiss  36929