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Theorem 1st2ndb 7441
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 = ⟨(1st𝐴), (2nd𝐴)⟩)

Proof of Theorem 1st2ndb
StepHypRef Expression
1 1st2nd2 7440 . 2 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
2 id 22 . . 3 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
3 fvex 6424 . . . 4 (1st𝐴) ∈ V
4 fvex 6424 . . . 4 (2nd𝐴) ∈ V
53, 4opelvv 5351 . . 3 ⟨(1st𝐴), (2nd𝐴)⟩ ∈ (V × V)
62, 5syl6eqel 2886 . 2 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ → 𝐴 ∈ (V × V))
71, 6impbii 201 1 (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  wcel 2157  Vcvv 3385  cop 4374   × cxp 5310  cfv 6101  1st c1st 7399  2nd c2nd 7400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-1st 7401  df-2nd 7402
This theorem is referenced by:  wlkcpr  26878  wlkeq  26883  opfv  29967  1stpreimas  30001  ovolval2lem  41603
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