ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opelres GIF version

Theorem opelres 4645
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⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))

Proof of Theorem opelres
StepHypRef Expression
1 df-res 4385 . . 3 (𝐶𝐷) = (𝐶 ∩ (𝐷 × V))
21eleq2i 2120 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)))
3 elin 3154 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∩ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)))
4 opelres.1 . . . 4 𝐵 ∈ V
5 opelxp 4402 . . . 4 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ (𝐴𝐷𝐵 ∈ V))
64, 5mpbiran2 859 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝐷 × V) ↔ 𝐴𝐷)
76anbi2i 438 . 2 ((⟨𝐴, 𝐵⟩ ∈ 𝐶 ∧ ⟨𝐴, 𝐵⟩ ∈ (𝐷 × V)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
82, 3, 73bitri 199 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝐶𝐴𝐷))
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102  wcel 1409  Vcvv 2574  cin 2944  cop 3406   × cxp 4371  cres 4375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-opab 3847  df-xp 4379  df-res 4385
This theorem is referenced by:  brres  4646  opelresg  4647  opres  4649  dmres  4660  elres  4674  relssres  4676  resiexg  4681  iss  4682  asymref  4738  ssrnres  4791  cnvresima  4838  ressn  4886  funssres  4970  fcnvres  5101
  Copyright terms: Public domain W3C validator