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Theorem 1st2ndb 6388
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 6387 . 2  |-  ( A  e.  ( _V  X.  _V )  ->  A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 id 21 . . 3  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3 fvex 5743 . . . 4  |-  ( 1st `  A )  e.  _V
4 fvex 5743 . . . 4  |-  ( 2nd `  A )  e.  _V
53, 4opelvv 4925 . . 3  |-  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  ( _V 
X.  _V )
62, 5syl6eqel 2525 . 2  |-  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  ->  A  e.  ( _V  X.  _V ) )
71, 6impbii 182 1  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   _Vcvv 2957   <.cop 3818    X. cxp 4877   ` cfv 5455   1stc1st 6348   2ndc2nd 6349
This theorem is referenced by:  opfv  24059  2wlkeq  28304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fv 5463  df-1st 6350  df-2nd 6351
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