MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  1st2ndb Structured version   Visualization version   GIF version

Theorem 1st2ndb 7962
Description: Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
Assertion
Ref Expression
1st2ndb (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)

Proof of Theorem 1st2ndb
StepHypRef Expression
1 1st2nd2 7961 . 2 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 id 22 . . 3 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
3 fvex 6856 . . . 4 (1st𝐴) ∈ V
4 fvex 6856 . . . 4 (2nd𝐴) ∈ V
53, 4opelvv 5673 . . 3 ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (V × V)
62, 5eqeltrdi 2846 . 2 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → 𝐴 ∈ (V × V))
71, 6impbii 208 1 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  Vcvv 3446  cop 4593   × cxp 5632  cfv 6497  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  wlkcpr  28580  wlkeq  28585  opfv  31564  1stpreimas  31622  ovolval2lem  44891
  Copyright terms: Public domain W3C validator