Theorem 1st2ndb 7956

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 7955	. 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 6830	. . . 4 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6830	. . . 4 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opelvv 5651	. . 3 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ (V × V)
6	2, 5	eqeltrdi 2839	. 2 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → 𝐴 ∈ (V × V))
7	1, 6	impbii 209	1 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⟨cop 4577   × cxp 5609  ‘cfv 6476  1^st c1st 7914  2^nd c2nd 7915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-iota 6432  df-fun 6478  df-fv 6484  df-1st 7916  df-2nd 7917

This theorem is referenced by:  wlkcpr  29602  wlkeq  29607  opfv  32618  1stpreimas  32679  ovolval2lem  46681  tposideq  48919  fuco22a  49382