Theorem 1st2ndb 8054

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 8053	. 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 6919	. . . 4 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6919	. . . 4 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opelvv 5725	. . 3 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ (V × V)
6	2, 5	eqeltrdi 2849	. 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 1540   ∈ wcel 2108  Vcvv 3480  ⟨cop 4632   × cxp 5683  ‘cfv 6561  1^st c1st 8012  2^nd c2nd 8013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fv 6569  df-1st 8014  df-2nd 8015

This theorem is referenced by:  wlkcpr  29647  wlkeq  29652  opfv  32654  1stpreimas  32715  ovolval2lem  46658  tposideq  48788  fuco22a  49045