Theorem opelres 4965

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 4688	. . 3 ⊢ (𝐶 ↾ 𝐷) = (𝐶 ∩ (𝐷 × V))
2	1	eleq2i 2272	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)))
3		elin 3356	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)))
4		opelres.1	. . . 4 ⊢ 𝐵 ∈ V
5		opelxp 4706	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ (𝐴 ∈ 𝐷 ∧ 𝐵 ∈ V))
6	4, 5	mpbiran2 944	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ 𝐴 ∈ 𝐷)
7	6	anbi2i 457	. 2 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))
8	2, 3, 7	3bitri 206	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ↾ 𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ 𝐴 ∈ 𝐷))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∈ wcel 2176  Vcvv 2772   ∩ cin 3165  ⟨cop 3636   × cxp 4674   ↾ cres 4678

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4107  df-xp 4682  df-res 4688

This theorem is referenced by:  brres  4966  opelresg  4967  opres  4969  dmres  4981  elres  4996  relssres  4998  resiexg  5005  iss  5006  restidsing  5016  asymref  5069  ssrnres  5126  cnvresima  5173  ressn  5224  funssres  5314  fcnvres  5461