Theorem 1st2ndb 8022

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 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Proof of Theorem 1st2ndb

Step	Hyp	Ref	Expression
1		1st2nd2 8021	. 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		id 23	. . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
3		fvex 6892	. . . 4 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6892	. . . 4 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opelvv 5699	. . 3 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ (V × V)
6	2, 5	eqeltrdi 2877	. 2 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → 𝐴 ∈ (V × V))
7	1, 6	impbii 212	1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ⟨cop 4597   × cxp 5657  ‘cfv 6534  1^st c1st 7980  2^nd c2nd 7981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6490  df-fun 6536  df-fv 6542  df-1st 7982  df-2nd 7983

This theorem is referenced by:  wlkcpr  29915  wlkeq  29920  opfv  32926  1stpreimas  32988  ovolval2lem  47244  tposideq  49546  fuco22a  50008