Theorem 1st2ndb 8011

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 8010	. 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 6897	. . . 4 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6897	. . . 4 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opelvv 5709	. . 3 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ (V × V)
6	2, 5	eqeltrdi 2835	. 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 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629   × cxp 5667  ‘cfv 6536  1^st c1st 7969  2^nd c2nd 7970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971  df-2nd 7972

This theorem is referenced by:  wlkcpr  29390  wlkeq  29395  opfv  32374  1stpreimas  32431  ovolval2lem  45913