Theorem 1st2ndb 7844

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 7843	. 2 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2		id 22	. . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
3		fvex 6769	. . . 4 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6769	. . . 4 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opelvv 5619	. . 3 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ (V × V)
6	2, 5	eqeltrdi 2847	. 2 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → 𝐴 ∈ (V × V))
7	1, 6	impbii 208	1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   × cxp 5578  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by:  wlkcpr  27898  wlkeq  27903  opfv  30883  1stpreimas  30940  ovolval2lem  44071