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Theorem 1st2ndb 6774
Description: Reconstruction of an ordered pair in terms of its components. (Contributed by NM, 25-Feb-2014.)
Assertion
Ref Expression
1st2ndb  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )

Proof of Theorem 1st2ndb
StepHypRef Expression
1 1st2nd2 6773 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 id 22 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3 fvex 5813 . . . 4  |-  ( 1st `  A )  e.  _V
4 fvex 5813 . . . 4  |-  ( 2nd `  A )  e.  _V
53, 4opelvv 4987 . . 3  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  ( _V 
X.  _V )
62, 5syl6eqel 2496 . 2  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  A  e.  ( _V  X.  _V ) )
71, 6impbii 188 1  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1403    e. wcel 1840   _Vcvv 3056   <.cop 3975    X. cxp 4938   ` cfv 5523   1stc1st 6734   2ndc2nd 6735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-iota 5487  df-fun 5525  df-fv 5531  df-1st 6736  df-2nd 6737
This theorem is referenced by:  2wlkeq  25006  opfv  27810  1stpreimas  27849
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